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[ ]

CONTENTS.

(A)

VOL. 199.

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page v

I. The Stability of a Spherical Nebula. By J. H. Jeans, B.A., Fellow of Trinity

College, and Isaac Newton Student in the University of Cambridge. Com¬
municated by Professor G. H. Darwin, F.R.S. . page 1

II. Continuous Electrical Calorimetry. By Hugh L. Callendar, F.R.S. , Quean

Professor of Physics at University College, London . 55

III. On the Capacity for Heat of Water between the Freezing and Boiling-Points,

together with a Determination of the Mechanical Equivalent of Heat in Terms
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Method of Calorimetry, performed in the Macdonald Physical Laboratory of
McGill University , Montreal. By Howard Turner Barnes, 31. A. Sc.,

D.Sc., Joule Student. Communicated by Professor H. L. Callendar,

F.R.S. . 149

IV. On a Throw-Testing Machine for Reversals of Mean Stress. By Professor
Osborne Reynolds, F.R.S., and J. H. Smith, M.Sc., Wh.Sc., Late Fellow

of Victoria University, 1851 Exhibition Scholar . 265

a 2

[ iv ]

V. The Mechanism of the Electric Arc. By (Mrs.) Hertha Ayrton. Communi¬

cated by Professor J. Perry, F.R.S. . page 299

VI. On Chemical Dynamics and Statics under the Influence of Light. By Meyer

Wilderman, Ph.D., B.Sc. (Oxo7i.). Communicated by Dr. Ludwig Mond,
F.R.S. . 337

AVI. Cycinogenesis in Plants. — Part II. The Great Millet, Sorghum vulgare. By

AVyndham E. Duxstan, M.A., F.R.S., Director of the Scientific a7id
Technical Department of the Imperial Institute, and Thomas A. Henry,
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A ll I. A Memoir on Integral Functions. By E. AV Barnes. M.A., Fellow of Trinity
College, Cambridge. Communicated by Professor A. E. Forsyth, Sc.D.,
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[ vi ]

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dishonour of the Society.

INDEX SLIP.

Jeans, J. H. — The Stability of a Spherical Nebula.

Phil. Trans., A, yoI. 199, 1902, pp. 1-53.

Meteoritic Hypothesis of Planetary Evolution.

Jeans, J. H. Phil. Trans., A, vol. 199, 1902, pp. 1-53.

Nebular Hypothesis.

Jeans. J. H. Phil. Trans., A, vol. 199, 1902, pp. 1-53.

Planetary Systems — Evolution of.

Jeans, J. H. Phil. Trans., A, vol. 199, 1902, pp. 1-53.

Rotating Mass of Gas — Stability and Equilibrium of.

Jeans, J. H. Phil. Trans., A, vol. 199, 1902, pp. 1-53.

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PHILOSOPHICAL TRANSACTIONS .

I. The Stability of a Spherical Nebula .

By J. H. Jeans, B.A. , Fellow of Trinity College, and Isaac Newton Student in the,

University of Cambridge .

Communicated by Professor G. H. Darwin, F.R.S.

Received June 15, — Read June 20, 1901. Revised February 28, 1902.

Introduction.

§ 1. The object of the present paper can lie best explained by referring to a sentence
which occurs in a paper by Professor G. H. Darwin. # This is as follows : —

“ The principal question involved in the nebular hypothesis seems to be the
stability of a rotating mass of gas ; but, unfortunately, this has remained up to now
an untouched field of mathematical research. We can only judge of probable results
from the investigations which have been made concerning the stability of a rotating
mass of liquid. ”

In so far as the two cases are parallel, the argument by analogy will, of course, be
valid enough, but the compressibility of a gas makes possible in the gaseous nebula a
whole series of vibrations which have no counterpart in a liquid, and no inference as
to the stability of these motions can be drawn from an examination of the behaviour
of a liquid. Thus, although there will be unstable vibrations in a rotating mass of
gas similar to those which are known to exist in a rotating liquid, it does not at all
follow that a rotating gas will become unstable, in the first place, through vibrations
which have a counterpart in a rotating liquid : it is at any rate conceivable that the
vibrations through which the gas first becomes unstable are vibrations in which the
compressibility of the gas plays so prominent a part, that no vibration of the kind
can occur in a liquid. If this is so, the conditions of the formation of planetary
systems will be widely different in the two cases.

With a view to answering the questions suggested by this argument, the present
paper attempts to examine in a direct manner the stability of a mass of gravitating
gas, and it wall be found that, on the whole, the results are not such as could have
been predicted by analogy from the results in the case of a gravitating liquid. The

* “ On the Mechanical Conditions of a Swarm of Meteorites, and on Theories of Cosmogony,” ‘ Phil.
Trans.,’ A, vol. 180, p. 1 (1888).

VOL. CXCIX. — A 312. B

31.7.02.

o

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

main point of difference between the two cases can be seen, almost without
mathematical analysis, as follows : —

§ 2. Speaking somewhat loosely, the stability or instability may be measured by
the resultant of several factors. In the case of an incompressible liquid we may sav
that gravitation tends to stability, and rotation to instability ; the liquid becomes
unstable as soon as the second factor preponderates over the first. The gravitational
tendency to stability arises in this case from the surface inequalities caused by the
displacement : matter is moved from a place of higher potential to a place of lower
potential, and in this way the gravitational potential energy is increased. As soon
as we pass to the consideration of a compressible gas the case is entirely different.

Suppose, to take the simplest case, that we are dealing with a single shell of
gravitating gas, bounded by spheres of radii r and r -fi dr, and initially in equilibrium
under its own gravitation, at a uniform density p0.

Suppose, now, that this gas is caused to undergo a tangential compression or
dilatation, such that the density is changed from

Po fo Po “h — p«S,;,

where p„ is a small quantity, and S„ is a spherical surface harmonic of order n.

It will readily be verified that there is a decrease in the gravitational energy of
amount

47rr3 (drf t -- pn - - f f S/ sin 0 dd d<b.
v ' (2 n + 1) J J

As this is essentially a positive quantity, we see that any tangential displacement
of a single shell will decrease the gravitational energy.

This example is sufficient to show that when the gas is compressible, the tendency
of gravitation may be towards instability. The gravitation of the surface inequalities
will as before tend towards stability, but when we are dealing with a gaseous nebula,
it is impossible to suppose that a discontinuity of density can occur such as would be
necessary if this tendency were to come into operation. Rotation as before will tend
to instability, and the factor which makes for stability will be the elasticity of
the gas.

We can now see that there is nothing inherently impossible, or even improbable,
in the supposition that for a gaseous nebula the symmetrical configuration may
become unstable even in the absence of rotation. The question which we shall
primarily attempt to answer is, whether or not this is, in point of fact, a possible
occurrence, and if so, under what circumstances it will take place. To investigate
this problem, it will be sufficient to consider the vibrations of a non-rotating nebula
about a configuration of spherical symmetry.

§ 3. Unfortunately, the stability of a gaseous nebula of finite size is not a subject

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

3

which lends itself well to mathematical treatment. The principal difficulty lies in
finding a system which shall satisfy the ordinarily assumed gas equations, and shall
at the same time give an adequate representation of the primitive nebula of
astronomy.

If we begin by supposing a nebula to consist of a gas which satisfies at every
point the ordinarily assumed gas equations, and to be free from the influence of all
external forces, then the only configuration of equilibrium is one which extends to an
infinite distance, and is such that the nebula contains an infinite mass of gas. The
only alternative is to suppose the gas to be totally devoid of thermal conductivity,
and in this case there is an equilibrium configuration which is of finite size and
involves only a finite mass of gas. But the assumption that a gas may be treated as
non-conducting finds no justification in nature. When we are dealing, as in the
present case, with changes extending through the course of thousands of years, we
cannot suppose the gas to be such a bad conductor of heat, that any configuration,
other than one of thermal equilibrium, may be regarded as permanent.

Professor Darwin has pointed out that a nebula which consists of a swarm of
meteorites may, under certain limitations, be treated as a gas of which the meteorites
are the “ molecules.”* In this quasi-gas the mean time of describing a free path must
be measured in days, rather than (as in the case of an actual gas) in units of
10~9 second. The process of equalisation of temperature will therefore be much
slower than in the case of an actual gas, and it is possible that the conduction of heat
may be so slow that it would be legitimate to regard adiabatic equilibrium as
permanent, t

Except for this the mathematical conditions are identical, whether we assume the
gaseous or meteoritic hypothesis. The present paper deals primarily with a nebula in
which the equilibrium is conductive, but it will be found possible from the results
arrived at, to obtain some insight into the behaviour of a nebula in which the
equilibrium is partially or wholly convective.

§ 4. Whether we suppose the thermal equilibrium of the gas to be conductive or
adiabatic, we are still met by the difficulty that the gas equations break down over
the outermost part of the nebula, through the density not being sufficiently great to
warrant the statistical methods of the kinetic theory. This difficulty could be avoided
by supposing that the nebula is of finite size, and that equilibrium is maintained by
a constant pressure applied to the outer surface of the nebula. If this pressure is so
great that the density of gas at the outer surface of the nebula is sufficiently large to
justify us in supposing that the gas equations are satisfied everywhere inside this
surface, then the difficulty in question will have been removed. On the other hand,
this pressure can only be produced in nature by the impact of matter, this matter

* G. H. Darwin, lor. tit., ante.
t Ibid., p. 64.

4

MR. J. H. JEANS OX THE STABILITY OF A SPHERICAL NEBULA.

consisting' either of molecules or meteorites, so that we are now called upon to take
account of the gravitational forces exerted upon the nebula by this matter. This
whole question is, however, deferred until a later stage ; for the present we turn to
the purely mathematical problem of finding the vibrations of a mass of gas which is in
equilibrium in a spherical configuration. We shall consider two distinct cases. In the
first, equilibrium is maintained by a constant pressure applied to the outer surface of
the nebula, this surface being of radius Xi j. In the second, the nebula extends to infinity,
and it is assumed that the ordinary gas equations are satisfied without limitation. We
suppose for the present that the gas is in thermal equilibrium throughout. It is not,
however, supposed that the gas is all at the same temperature ; to make the question
more general, and to give a closer resemblance to the state of things which may be
supposed to exist in nature, it will be supposed that the gas is collected round a solid
spherical core of radius H0, and the temperature will be supposed to fall off as we
recede from this core to the surface, the equation of conduction of heat being satisfied
at every point. We shall also suppose that the gas is acted upon by an external
system of forces, this system being, like the nebula, spherically symmetrical. The
reason for these generalisations will be seen later; it will at any time be possible to
pass to less general cases.

The Criterion of Stability.

The Principal Vibrations of a Spherical Nebula.

§ 5. We shall take the point about which the nebula is symmetrical as origin. It
will be convenient to use rectangular co-ordinates x, y, z, in conjunction with polar
co-ordinates r , 0, <f>. W e shall imagine the nebula to undergo a small continuous
displacement ; let the components of this be y, £, when referred to rectangular
co-ordinates, and u, r v, nv sin 0 when referred to polars. Thus the point initially at

IP 2 or r, 0, $

is found after displacement at

a; + £ y + y, z + £ or r + u, o + V, 4> + U\

The cubical dilatation of this displacement will be denoted by A. so that

°ii fi

6y 8:

k-(hV) +

cr x

1_ 0_

sin 0 c6

(r sin 0) +

CIO

dfi

MR, J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

5

In general we shall denote the density by p, pressure by txt, temperature by T,
toted potential by V, coefficient of conduction of heat by k, and the yas constant by X,
the last of these being given by the equation

trr = XT p . ( 1 ).

In the equilibrium configuration each of the quantities just defined is a function of
r only.

If c is any one of these quantities, we shall denote the

Value of c in the equilibrium configuration, evaluated at x, y, 2, by c0.

„ „ displaced „ „ „ „ c0 + c.

,, ,, ,, ■> , ■>•> ■ 1 d- ■> y — t- y 1 • d~ C by ^ 0 d- ^ 1 •

The quantities c0, c, <q are, of course, not independent. Since c0 + cl is the same
function of x -f- £ y y, 2 -f C, as is c0 fi- c' of x, y, z, we have, as far as t lie first
order of small quantities,

c\) d~ — <0 d~ c d~

dc,

c Z~0

^ dx

1 1 y

+ 7J¥ + ?

hc0

dz’

or, since cv is a function of r only,

clcn

0 - c' + u A

§ 6. From the equation of continuity we have at once

Pi — ~ Po A

(3).

Since X remains the same throughout the motion of any given element of the gas,

J

*i = 0. . (4).

Hence, from equation (1),

a*u d~ ~~ *0 (To d- Ti) (pu + Pi),

giving as the value of vr,

"i — *0 (T,Po -F I’upi) — \)po (d\ — AT0)

(5).

So long as we confine our attention to a single element of the gas, the coefficient of
conduction of heat is proportional to the square root of the temperature, and is

6

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

independent of the density.* We therefore have, as far as the first order of small
quantities,

*1

Lastly V', regarded as the difference between V0 + V' and V0, is seen to be the
potential of a volume-distribution of matter of density p', to which must be added :

(i.) The potential of a surface-distribution over the sphere r = R0, the surface
density being

- [■ u(Po - o-0)]r = Ko,

where <x0 is the mean density of the core, and

(ii.) The potential of a surface-distribution over the sphere r = Rl5 the surface
density being

[«(P0 ~ 0‘l)]r = B1,

where oq is the density of the medium (if any) outside the nebula.

§ 7. We are now in a position to handle the equations of motion, and of conduction
of heat. For the element which, in the undisturbed state, is at x, y, z, the equations
of motion are three of the type

0^f _ _0

dt~ dx

(V0 + W) -

1 0
(Po + P)

+ cfi) .

(7).

Transforming to polar co-ordinates, these equations are equivalent to

9%

ot“

|(V„ + V')-

(po + p')

(CT0 + *0 •

(8).

0% _ 1 0V' 1_ dm'

dt2 r d9 p0r d6

(9)

. . dho 1 0V' 1 dm'

r Sill U -TTT = 7—7. w - ; — -

ct~ r sm 6 c</> p0r sm 6 cep

As an equation of equilibrium, we have

<Wp _ 1 0OTq _

dr p0 dr ~~

(10).

(11).

and with the help of this, equation (8) reduces to

dru 0V' 1 dm' p' 0wo
dt~ dr p0 dr p0~ dr ‘

as far as the first order of small quantities.

* Boltzmann, ‘ Vorlesungen liber Gastheorie,’ vol. 1, § 13.

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

7

Let us write

X = v ~ OTVpo • •

so that

0y 0V' 1 0cfi ro' dp0

dr dr p0 dr p0° dr ’

then equation (12) becomes

_i JL l ' _ / ?W\

dt2 dr p02 [f dr m dr ) ’

(13),

and, by the use of equation (2), this is seen to be equivalent to

d2u _dx . 1 / 3®o 0Po\

dt2 ~ dr + Po2 \Pl dr ^ dr) ' *

Equations (9) and (10) now take the simple forms,

02r 1 0^ 0% 1 0y

0i2 ?’2 d0 dt 2 r2 sin2 # 0c/> ’

(14).

From these last two equations, we obtain at once

1 0
r2 sin 0 d0

+

02

X

r2 sin2 6 dcf r

or, what is the same thing,

02_
dt 2

= V2x - is

dr

7 *

9%

dr

(15).

§ 8. The equation of conduction of heat is, as far as the first order of small
quantities,

in which p, k, T stand for Po + p , k0 -j- k , t0 + t respectively. The notation is that
of Kirchhoff ; the equation may either be written down from first principles, or
regarded as a simplified form of Kirchhoff’s general equation.*

Since there is thermal equilibrium in the undisturbed configuration,

_0_

dx

3T0\
0 dx J

If 0To\

02 r° 02 j

= 0.

(17).

Kirchhoff, ‘Vorlesungen liber die Theorie der Warme, p. 118.

8 ME. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

Hence equation (16) reduces to the form

M Ul . n _ Ml f ST'\ , 3 / 3T'\ , JL f ^

M dt + C° dt ” />o L 3^ V 0 &/ + ( *° 3y / + 3* r° a?

+ s('/i) + iu'f HAH vU ■ <l8>-

a / , st(

dz \ c

Since k0, T0 are functions of r only, the bracket on the right-hand side of this last
equation again reduces to

a*o cl T-rorry , S/C cTn /r72rT

+ ar + ,cV'io

0r 0r 1 ~° v A 1 dr dr 1 " v .

and, cleared of accented symbols by the use of equation (2), this takes the form

a*o ST, T7r’rr i *afo i r7orr

a: a: + *ov'Ti + 757 -jT. + «iv'To

— U

or 01
3 fdnn 0T,

0 \ , yo ( cTn \ , a«o y;y

' °X Ur / + dr 0

dr \ dr d ?

duj a_yo0To , . anj _ ar,

" a.. 1 0 , v, ~r Ko 77.9. r

or [ dr dr "'° d/
Now equation (17) can be written in the form

0 t~~2 „ .

K0

(20).

'N

O,co °t0

3 r hr T ' 1,1

+ k0 V-’T0 = 0.

(21),

whence, by differentiation with respect to r.

0

d/cn

3^ /cy0 0Tq\ , y2rp , — -o yorp _ ~

dr\dr dr I + u dr 0 + 0r V io “ U-

(22).

With the help of equation (22), the bracket in the second line of (20) reduces to

2r0 01V
r~ dr ’

while, with the help of (21), that in the third line becomes

9 is 0T

_ ~__S) UJ-o #

r or

Again, if we substitute for k] the value found for it in equation (6), the two last
terms in the first line of (20) can be transformed as follows :

a*i 3Tn a / Tj \ 0t„ t, ra*0 8t0

a7 a7 + **v T» = *0 a- ( wj e7 + Tt; 1 07 a7 + *0 V'T«

and the last bracket vanishes by equation (21).

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

Collecting results, and substituting for px from equation (3), we find that equation
(18) takes the form

0A 3T,

Mr» st + ar

i ra^ST , a/T, \ai’o

+ ^oVil + Ku ' - '

dr dr

0 dr \2Tj dr

0Tn/2 u 4 die \

- -j- + V-u
r or

_i9 ‘

dr \ r*

(23)*

§ 9. In addition to the volume-equations which have just been found, there are
certain boundary conditions which must be satisfied. These are as follows :

(i.) The pressure must remain constant at the outer surface, so that we must
have

OlJ-E, = 0.

(ii.) The temperature must remain unaltered at r = It0, or else the flow of
temperature across the surface r = 11 0 must remain ml. These two suppositions
require respectively

[Ti],=Ko = 0, or

(iii.) A similar temperature condition must be satisfied at r ~ fq.

(iv.) The kinematical and dynamical boundary conditions at the surface r — I1(J
must be satisfied. These express that the normal velocities shall be continuous at
this surface, and that the motion of the rigid core shall be such as would be caused
by the forces acting upon it from the gas.

§ 10. Equations (14), (15) and (23) give the rates of change in u, A and 1\ in terms
of these quantities. Hence these equations enable us theoretically to trace the
changes in u , A and T1; starting from any arbitrary values of u, A, Tx, du/dt and
dA/dt, which are such as to satisfy the boundary conditions.

Imagine initial values of u, A, T’j, du/dt and dA/dt, in which the latitude and
longitude enter only through the factor S,„ where S,, is any spherical harmonic of
order n. Then it can be shown that the solution through all time (so long as the
squares of the displacement may be neglected) is such that the latitude and longitude
enter only through the factor S„. For, assuming a solution of this form, the value of
V' found in § 6 will contain S„ as a factor, as will also /q, cTT 1 5 77) (equations 3, 5, 2)
and y (equation 13). The same is true of V:y, V'T1 and Vhq since

tZTj

dr

r=Bn

n(n + 1 )/(?’) 1 o

* Sections 5-8 were re-written in November, 1901. I take this opportunity of expressing my thanks
to the referee for the care and trouble which he has bestowed upon my paper. To him I am indebted for
several improvements in these four sections, in particular for the present form of equation (23), and also
for the removal of a serious inaccuracy from my original equations.

VOL. CXC1X. - A. 0

10

MR. J. II. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

where f(r) is any function of r. It therefore appears that every term in equations (1 1),
(15) and (23) will contain S;, as a factor. Dividing out by this factor, we are left with
equations which do not involve 0 and </> , and this verifies our statement.

§ 11. It therefore follows that there are principal vibrations* in which u, A and T,
are of the form

u — AS . (24),

A = BS„e^

Tj = CS„e‘A

in which A, B, C are functions of r only. The relations between A, B, C and p must
be found from the equations (14), (15), (23), and the boundary conditions.

The value of p' for the vibration just specified is

u

dPo

dr

Ap0 A u

dPo\
dr )

dPo

dr

-fi Bp0 )

We shall in future drop all zero suffixes, there being no longer any danger of
confusion. Calculating V' after the manner explained in § 6, we find (cf Thomson
and Tait, ‘Nat. Phil./ § 542),

V' = VS.es*,

where

V =

47 r

(in + 1)

^ { - j k (A % + Bp) dr - [A (p - <70)

4t ri'“

+ (2?l+ 1)

r

cf

A (p — oq)

r=ll,

. . . (27).

We have further, by equations (2) and (5),

Cl TV

= oT, — U

UTS

dr

= \P (C - BT) - A S;;c'C,

and hence we obtain (equation 13)

X = FS.eC,

where

F = V - X (C - BT) + A ly . (28).

v ' p dr

Substituting the assumed solutions for u, A and If, and the corresponding values
for y, pl5 btjl, in equations (14), (15) and (23), and dividing throughout by the factor
we find the relations

* In order to avoid circumlocution, we shall find it convenient to use the terms “principal co-ordinate ”
and “ principal vibration,” although we are ignorant as to whether the nebula is stable or unstable. It
will ultimately be found that we only apply our results to nebulae which are either stable or in the limiting
state of neutral equilibrium.

ME. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

11

dV

dr

p dr

— ^ (C - BT)

p dr v 7

»(” + i) F

(29).

ipp(MpB+C,C) = ~ +

k j d

r* [ dr

o dC'

V~ dr

— n (n + 1) C

d

+ K7r

'_C\ JT
2T/rfr

k^TJ 1

rfr j r dr \ dr )

4 dA _ n(n + 1) - 2
r dr “ /'2

The boundary-equations found in § 9 reduce to the following : —
(i) [C - BT],=I?1 = 0 .

= 0 . (33),

-'■=«o

(iii) Equations similar to (33) at r — Pi^ . (34),

(iv) (A)r=Ro = 0, when n is different from unity, or a more complex equation in the

case of n = 1 . (35).

(ii) C,=no = 0 or

clC

dr

§ 12. From the manner in which the analysis has been conducted, it will be clear
that every principal vibration must either be one of the class just investigated, or
else a vibration such that u, A, and T vanish everywhere.

For the latter class of vibration there are no forces of restitution. Thus the
frequency of vibration is zero, and the motion consists of the flow of the gas in closed
circuits, this flow being entirely tangential, and the gas behaving like an incom¬
pressible fluid. Obviously these steady currents are of no importance in connection
with the question of stability or instability.

Discussion of the Frequency Equation.

§ 13. Returning to the class of vibrations in which u, A, and T do not all vanish,
we have seen that the frequency equation is found by the elimination of F, A, B, and
C from equations (28) to (35). Now q> only enters into three of these equations :
namely (31), in which it enters through the factor ip, and (29) and (30), in which it
enters through the factor — p~ or (ip)2. Regarding ip, A, B, C, and F as unknowns,
it will be seen that the coefficients which occur in equations (28) to (35) are all real.
The four volume equations enable us to determine A, B, C, and F except for certain

12

MR. J. H. JEANS ON THE STABILITY' OF A SPHERICAL NEBULA.

constants of integration, and the values of these quantities will be wholly real if irp is
real. The boundary-equations enable us to determine the constants of integration
and also provide an equation for ip. Every term in these equations will be real if ip
is real. Hence the frequency equation can be written in the form

f(ip) = 0,

where f(x) is a function of x in which all the coefficients are real, these coefficients
being functions of n and of the quantities which determine the equilibrium configura¬
tion of the nebula.

It follows that the complex roots of ip will occur in pairs of the form

ip = 7 ± i S,

where y and 8 are both real. There may also be roots for which ip is purely real, so
that 8 = 0, and y exists alone.

The vibration corresponding to any root is stable or unstable according as y is
negative or positive.

If the equilibrium configuration of the nebula changes in any continuous manner,
so as always to remain an equilibrium configuration, the values of ip will also change
in a continuous manner, and for physical reasons these values can never become
infinite. Hence, if the configuration of the nebula changes from one of stability to
one of instability, it must do so by passing through a configuration in which there is
a vibration for which y = 0.

§ 14. For the present we shall not discuss the actual stability or instability of any
configuration, but shall examine under what circumstances a transition from stability
to instability can occur.

We therefore proceed to search for configurations in which there are vibrations
such that y — 0. Now for such a vibration we have either a root of the frequency
equation p = 0, or else a pair of roots of the form ip — ffi i 8.

In the latter case the corresponding vibration is one in which a dissipation of energy
does not occur. A necessary condition for such a vibration is that no conduction of
heat shall take place. Hence both sides of the equation of conduction of heat
(equation 31) must vanish. Excluding adiabatic motion (represented by the
vanishing of the factor MpB -f- C*.C), this condition compels us to take

P

= 0

together witl

chc dC k [ d / 0 dC\ . i \ ru i

dr d i

dr I

a

j

d/C\ dT

K

dr \2T di

— K

dT f

Id/

4 dA n (n + 1) — 2 1

dr\

p dr \

. dr J

i

1

VJ

= 0

(3G).

MR, J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

13

Thus vibrations for which y = 0, if they exist, must satisfy equations (32) to (36),
and also equations (29) and (30), in which p is put equal to zero, and equation (28).

The case of n = 0 will be considered later (§ 28). Excluding this for the present,
we find that putting p = 0 in (30) leads to

F - 0 . . .

... (37).

Equation (29) now reduces to

B (1y 4- X q- (C — BT) = 0 . . . .

dr dr

DO

or, replacing — by its value XTp,

b4,(xt) + x|c=o ....

. . . . (39).

Equation (28) becomes

Y — X (C — BT) — — y = 0 . . .

. . . . (40),

and the elimination of C — BT from this equation and (38) leads to the equation

l/p V = - (A y + Bp

dr \ dr '

\ 1 f/w
p dr

Substituting for Y from equation (27), this becomes
4,r r|V,,A'i + P> p)r"*"-dr +

(41).

(2 n + 1 )r»+! Ljb„ v
47T?-n

"l / c

A (p - tr,,) rn+z

>=Ro

nM — \

(2)1 +1) [Jr

§ 15. With a view to transforming this equation, let us consider the equation

4-7T f f' t ,o 7 , Tr 1 . 4c7T),n

| Jrw+3 dr + K0 1 +

(42).

(2 n + 1) rn+1 [ J

(2a + 1)

h dr + K, ) = L (43).

1

in which J and L are any functions of r, and K0, K1 are constants. If we multiply
by rn+l, and differentiate with respect to r, we obtain, after some simplification,

+ =|(Lr-T .

• • (44),

while by multiplying (43) by r " and differentiating, we obtain in a similar way

d

-m:

YhA 1 Jr”+2 dr + K

di

7 (Lr-") . . . .

(45).

14

MR. ,J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

Divide (44) by r2>‘ and differentiate with respect to r, then

47rJ

or, writing £ for Lr, and simplifying

n (n + 1 )

— 47T?'J

(«)>

(V).

and this same equation could have been deduced from (45) instead of (44).

Equation (47) is more general than (43) since the two constants K0, K, have
disappeared. In fact equation (47), being a differential equation of the second order,
will contain two arbitrary constants in its solution, and these correspond to the two
missing constants K0 and K1. We can, however, determine K0, K, in terms of
these two arbitrary constants, and if these constants are chosen so as to give the
right values for K0, KL, the solution of (47) will be equivalent to the original
equation (43).

To determine Kn, K,, put r = R1 in (44) and we obtain

and similarly from (45)

inK,

~l d
dr

(48),

47tKa = —

di

:(&- ("+,))

i'=Bn

(«)•

Hence we see that equation (43) is exactly equivalent to the three equations
(47), (48), and (49).

§ 16. Comparing (42) with (43), it appears that (42) is exactly equivalent to the
following equations : —

+ Bp )

/

T dm j dp

p dr / dr

(50). “

n (n + 1)

, o
0

- 4771' (A + Bp i

(51).

4ttK, =

Cyll\

~A(p — a j)

i —

J

’•=R1

rn~ 1

47tK0— —

,,3 m + 3

d_

dr

:{€r {n+1)) = A (p — «x 0) i

J>-=Ro

a %n + 2

,=nu

(52) .

(53) .

The right-hand member of (51) is equal to

ME. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

15

so that if we introduce a new quantity u, defined by

o o dp / dm

u = 2rrpr-~/-

equation (51) may be written in the form

{n (n + 1) — 2 u) $

(54).

(55).

The solution of this will be of the form

f = K.'/j (>') + J'-.'/'. (''■)

(56),

in which El5 Eo are constants of integration. We have, from the definition of

Af + B', = w = sW‘<r> + E*W> .... (57),

and the elimination of B from this equation and (39) gives

fiv C = !<XT>{A! -

(58).

If we imagine this value for C substituted in equation (36), we shall have a
differential equation of the second order for A. The solution of this will be of the
form

A

— + Eo J2(r) + E3/3 (r) + Ej./4(r)

(59),

in which E3 and E^ are the new constants of integration. From this value of A we
can deduce the values of B and C (equations (57) and (58)) without introducing any
further constants of integration.

Turning to the boundary conditions, we now find that there are six boundary-
equations to be satisfied (equations (32), (33), (34), (35), (52), (53)) and only three
arbitrary constants at our disposal, namely, the ratios of the four E’s. If we
eliminate these E’s we shall be left with three equations to determine the configura¬
tion of the nebula at which instability sets in, and these equations will iii general be
inconsistent.

§ 17. In order to put the right interpretation upon this result, it will be necessary
to return to the general equations of free vibrations found in § 12.

Il we eliminate F from equations (29) and (30), we obtain

16

MR. J. II. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

*‘{a + ^ i» £ (’'3B) - i (,'3A)I = 7*c + £ (XT> B • ■ <G0)’

w

liile equation (30) may, with the help of (28), be written in the form

V = f/r .

in which £ is now defined by

A dr< i

- =X(C-BT)- - +

r

p civ n (n 1)

d

• (01),

(02).

Substituting for V from equation (27), and treating the equation so formed in the
manner explained in § 15, we find, as the equivalent of equation (61),

(i.) A volume equation, analogous in form to (51), namely,

dr*

• (63).

(ii.) Two boundary equations analogous in form to (52) and (53). . . (64), (65).

Thus the equations found in § 11 may be replaced by

(a) Thi •ee volume equations, namely, equations (60), (63), and (31).

(f3) Six boundary equations, namely, equations (32), (33), (34), (35), (64), (65).

We may conduct the elimination of B and C from the three equations (a) in a
symbolic manner as follows : — -

Let D„ be a symbol which is used to denote any linear differential operator of
order n, the differentiations being with respect to r. The symbol has reference
solely to the order of the highest differential coefficient which occurs, and must in no
case have reference to any particular differential operator. Thus we write D;,
indiscriminately for every operator of* the form

a»- 1

f,L (r) 3 ,.u /«-i (r) 3,.»-i + • • •

The laws governing the manipulation of this symbol are as follows

(i.) — I )„<jf> = ILc/j,

(ii.) 1) „<J) + D = 1 )„(() (n > m),
(hi-) = Dw+ll</».

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 17

It must be particularly noticed that in general

D;i<£ — D„<£ = D „<f>.

Corresponding, however, to any two specified operators of order n, say (D„), and
(D„)o, it will always be possible to find two functions of r, say a and b, such that

a (D„)1 <f> — b (D„)2 (f) = D,;_,c£ . (6G).

In terms of this operator, the three equations (a) (p. 16) may be written in the
following forms :

P~ (Do A -j- DXB) + D0B + D0C = 0 . (67),

D3A -j- D2B + D2C + p3 (DSA + D2B) — 0 . (68),

lP (D0B + D0C) + DoC -)- D2A — 0 . (69).

Now D„ is commutative with regard to functions of r, and is of course commutative
with regard to p. This enables us to eliminate B and C from the above equations.

To make this clearer, consider a simple case, say the pair of equations

D2A = D;iB . (70).

D1A = p2BmB . (71).

If we operate on (71) with d/dr, we get an equation of the form

D.A = p3Dm+1B,

and from this and equation (70), we can, with the help ot the property expressed in
equation (66), deduce an equation of the form

DjA = D„B + jAD„;+1B.

From this and equation (71) we can in a similar way obtain an equation of the form

D0A — DJ3 + p~ DW+1B.

We may regard this as an equation giving A, and substitute for A in (71). In this
way we obtain

D„+1B -fi p3D,ft+2B =0 . (72),

and the elimination of A has been effected.

It will be clear that throughout this elimination we have followed a method which
would have been successful in eliminating A if d/dr had been regarded as a mere
multiplier. The result of the elimination is accordingly exactly the same as might
have been obtained directly from the original equations (70) and (71), by regarding
the D’s as multipliers and eliminating according to the ordinary laws of algebra.

VOL. CXCIX. — A.

D

18

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

It will now be apparent that we can eliminate any two of the three unknowns, A,
B, and C, from equations (67)-(69) by this method. The differential equation satisfied
by the remaining unknown (say A) will be

where, symbolica

A =

AA = 0 .

p2D2, p*Di A D0, D0
p2D3 A D2, + d2, d3

D2, ipD 0, ip D0 + Do

(7

3),

(74).

We may expand this determinant according to the rules already laid down for the
manipulation of the D’s, and so obtain

A — ip° D4 + a4D6 A ^3D4 + ,P3D 6 A A D4 . • • • (75).

§ 18. We can now see the explanation of the difficulty which occurred in § 16.
The occurrence of the term DG in A points to a differential equation of the sixth
order, which is satisfied by any one of the quantities A, B, or C in the general case,
in which p does not vanish. As soon, however, as p is put equal to zero, the
expression for A reduces to D4, and the differential equation is one of the fourth order
only. It therefore appears that by putting p — 0 before solving the differential
equations, the order of these equations is reduced automatically, and two solutions
are entirely lost from sight.

These two last solutions, it is easy to see, are solutions which do not approximate
to a definite limit, when p> approximates to zero. The remaining four solutions will
approximate to the same forms as would be obtained by putting p = 0 before solving
the differential equations. Thus, instead of equation (59), we must write the complete
limiting solution for A in the form

L' A — Elt/j (r) A E3 f2 (r) A E3/3 (r) + E4/4 (r) A E5/5 (r.

p= 0

_L It f la

\ A-

* I have not found it possible to investigate the form of these two last solutions in the general ease, but
it is easy to examine the nature of the solutions at infinity, when the nebula extends to infinity, and this
enables us to form some idea as to the general nature of the solutions. Suppose that at infinity we have

j , 1 dTjr dX

r=x X p dr dr

± a2r~*

in which a is real, then it can be shown that A = (r, p) A', &c., in which A', B’, C', are functions of

r only, and

(r, p) = E5 cos (2 Jan (n + 1) r~si2/isp) + EG sin (2 Jan (n + 1 )r~sl2/isp)

when the negative sign is taken in the above ambiguity, the circular functions being replaced by hyperbolic
functions when the positive sign is taken. The value of ip is wholly real when squares of ip may be
neglected (c/. § 13).

ME, J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

19

If we deduce the values of B and C from the solution (76), and substitute in the six
boundary equations the values so obtained, we shall be left with six linear and
homogeneous equations between the six E’s. Eliminating the six E’s, we have a
single relation between n, the constants of the nebula and p. Now it will be seen
that it will always be possible to pass to the limit p = 0 in this equation, since this
amounts only to finding the ratio of the values of f-a or f6 at the two boundaries.
The equation obtained in this manner will give us a knowledge of the configurations
at which a change from stability to instability can take place.

§ 19. It therefore appears that it is not sufficient to consider vibrations of frequency
p — 0 as represented by positions of “limiting equilibrium.” The method of
PoincarB# for determining points of transition from stability to instability is not
sufficiently powerful for the present problem ; indeed it appears that it is liable to
break down whenever there are boundary-equations to be satisfied.!

It is of interest to notice that this complication is not (as might at first sight be
suspected) a consequence of our having taken thermal conductivity into account.
For we can put C = 0 and remove the equation of conduction of heat without causing
any change in our argument, except that the right-hand member in equation (74) must
be replaced by a determinant consisting only of the minor of the bottom right-hand
member in the present determinant. The value of A is now

A — p~ D4 + D2,

and the number of boundary- equations is of course reduced from six to four. Thus
an exactly similar situation presents itself, although we are now dealing with
a strictly conservative system.

The consequences of this result are more wide-reaching than would appear from
the present problem, inasmuch as all problems of finding adjacent configurations of
equilibrium are affected. For instance, it appears that an equilibrium theory of tides
is meaningless except in very special cases ( e.g ., when the elements of the fluid in
which the tide is raised are physically indistinguishable).

If we attempt to calculate by the ordinary methods the tide raised in a mass of compressible fluid by
a small tide-generating potential, we reach a number of equations which are (except in special cases)
contradictory. To take a simple case, suppose we have a planet of radius E0 covered by an ocean of
radius Rj, the whole being surrounded by an atmosphere which maintains a constant pressure n at the
surface of the ocean. Let the law of compressibility be w = op, where c varies from layer to layer of the
ocean. Let the tide generating potential be a0rn S,t. Then the equations of this paper will hold if we
write p = 0, C = 0, ignore the equation of conduction of heat, replace AT everywhere by c, and include in

* “ Sur l’Equilibre d’une Masse fluide . . . ‘ Acta. Math.,’ 7, p. 259.

t There is not, of course, a flaw in Poincare’s analysis, but he works on the supposition that the
potential-function is a holomorphic function of the principal co-ordinates, and this supposition excludes a
case like the present one.

D 2

20

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

Y a term a0rn. Equation (39) gives (except in the special case of c = constant), B = 0. Equations (50)
and (51) remain unaltered, and give a solution of the form

A = Ei/i (r) + Eo/2(?-).

Now we must have A = 0 when r = R0, and this determines the ratio Ei/E2. Also equation (49) must be
satisfied, and this leads to a second and different value for Ei/E2.

A second example, of less interest but greater simplicity, will perhaps help to elucidate the matter.
Imagine a non-gravitating medium in equilibrium under no forces inside a rigid boundary. Let the law
connecting pressure and density for any particle be w = up, where k varies from particle to particle. In
equilibrium vr has a constant value ~0. Suppose now that we attempt to find an adjacent configuration
which is one of equilibrium under a small disturbing potential Y. The general equations of equilibrium
are three of the form

dV _ 1 dvr

do: p dx

If the position of equilibrium only varies slightly from the initial position, dwfdx will be a small quantity
of the first order, so that (to the first order of small quantities) p may be replaced by its equilibrium value
vt0/k. We now have

dvr _ vtq dV
dx k dx ’

and therefore, since w is a single -valued function of position,

f 1 dV , A ,. x

" .

the integral being taken along any closed path. Since Y and k are absolutely at our disposal, this
equation is, in general, self contradictory. What we have proved is that there will only be an “ adjacent ”
configuration of equilibrium under a potential Y if V is a single valued function of k, a condition which
will not in general be satisfied by arbitrary values of Y and k.

It is not difficult to see the physical interpretation of this last result. There were initially an infinite
number of equilibrium positions, and therefore an infinite number of vibrations of frequency p = 0. To
arrive at the configuration of equilibrium under the disturbing force we must imagine vibrations of
frequency p = 0 to take place until equation (i.) is satisfied; the disturbed configuration will then differ
only slightly from the configuration of equilibrium. For instance, if the disturbing field of force consists
of a small vertical force g, the fluid must be supposed to arrange itself in horizontal layers of equal density,
before we attempt to find the disturbed configuration.

The interpretation of the result obtained in the first instance is similar, but more difficult. Consider
a linear series of equilibrium configurations, obtained by the variation of some parameter a, such that the
spherical configuration of our example is given by a = 0. The other configurations are not symmetrical,
the asymmetry being maintained, if necessary, by an external field of force. Every degree of freedom in
the configuration a = 0 must have its counterpart in the configurations in which a is different from zero.
In particular, the principal vibrations of § 12, in which (for the configuration a = 0) the dilatation, normal
displacement, and temperature-increase all vanish, must have counterparts for all values of a. But when
a is different from zero, the above three quantities cannot be supposed to all vanish. In general, therefore,
these degrees of freedom provide solutions of the volume-equations, and these solutions contribute to the
boundary-equations. In the special case of a = 0, these solutions do not affect the boundary-equations at
all, so that to rectify the boundary-equations we must, so to speak, take an infinite amount of these
solutions. In other words, the complete vibration of frequency p = 0 becomes identical with one of the
vibrations of § 12, in which u, A, and Tx all vanish.

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

21

An Isothermal Nebula.

§ 20. Let us now examine the form assumed by our equations in the simple case in
which X and T are the same at all points of the nebula. We find that, considering
only the equations for the case of p — 0, equation (39) reduces to

C = 0 . (77),

and, in virtue of this simplification, the equation of conduction of heat (36), and the
two thermal boundary conditions (33 and 34) are satisfied identically. We are left
with equation (55) to be satisfied throughout the gas, and equations (32), (35), (52),
and (53) to be satisfied at the boundaries.

The solution of equation (55) is given in equation (56). Now we must satisfy
equation (32) by taking B = 0 at r = Rx, and this, by equation (50), gives the value
of A at r = Rj in terms of E2 and E2. Hence equation (52) reduces to a homo¬
geneous linear equation between E: and E3.

When n is different from unity, we satisfy equation (35) by taking A = 0 at
r = R0, and this reduces equation (53) to a homogeneous linear equation between
E: and E2.

When n — 1, equation (35) reduces to a linear equation between (A)r = Ro, Ex and E3.
Equation (53) is a second equation of the same form, and the elimination of (A),. = Eo
from these two equations leads to a homogeneous linear equation between E: and E2.

Thus, in either case, we see that the whole system of equations reduces to a pair
of homogeneous linear equations between E: and E2. The elimination of these
quantities leaves us with a single equation between n and the constants of the
nebula.

We can, therefore, satisfy all the equations for a vibration of frequency p — 0 by
imposing a single relation upon the constants of the nebula. The unknown solutions
which are multiplied by E5 and E6 have not been taken into account at all, but since
the condition that there shall be a vibration of frequency p = 0 must of necessity
reduce to a single equation, it will be clear that if these solutions had been taken
into account, we should have found it necessary to take E5 = E6 = 0.

Thus, in the case which we are now considering, a vibration of frequency p — 0
is equivalent to a configuration of limiting equilibrium. It is not hard to see that
this results from the fact that the particles of which the nebula is composed are
physically indistinguishable. This very fact, however, introduces a further complica¬
tion into the question. It will be noticed that, although the value of £ has been
found at every point of the nebula, it is impossible to determine the separate values
of A and B. On the other hand, the physical vibration must have a definite limiting
form when p — 0. Now it is easy to see that a motion of the gas in which
£ vanishes at every point of the gas, and in which A and B vanish separately at the

22

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

boundary, will, in every configuration of the gas, satisfy our equations with rp = 0.
Such a motion, in fact, simply leads to a configuration which is physically
indistinguishable from the initial configuration, and in which the potential energy
remains unaltered. The motion which we have found from our equations is the sum
of a motion of this kind, and a true limiting vibration. It is impossible to separate
the two motions, except by considering vibrations of frequency different from zero,
but fortunately the question is not one of any importance.

§ 21. Let us now attempt to form the final equation in some cases of interest. The
equations of an isothermal nebula at rest under its own gravitation have been
discussed by Professor Darwin. # Our function u (equation 54) is given, in the case
in which the nebula is isothermal, by the equation

u =

27 rpr2
XT ‘

(78).

and it will be seen that this is the same as the u of Professor Darwin’s paper. It
appears that in general u cannot be expressed as a function of r in finite terms, but a
table of numerical values of u is given, t The value of u approximates asymptotically
to unity at infinity, so that at infinity p varies as r~2. Darwin’s nebula extends
from r — 0 to r = oo , but it is obvious that we may, without disturbing the
equilibrium, replace that part of the nebula which extends from r = 0 to r = R0 by
a solid core of mass equal to that of the gas which it replaces. We may also remove
that part of die nebula which extends from r — R1 to r — °° , if we suppose a pressure
to act upon the surface r = P: of amount equal to the pressure of the gas at this
surface. We may suppose the medium outside this surface to be of any kind we
please, but as it lias already been pointed out that the pressure can, in nature, only be
maintained by the impact of matter, we shall suppose that this matter is of a density cr
which is continuous with the density p of the nebula at the surface of separation.
We may now write equation (52) in the simple form

(79).

We have, up to the present, supposed the nebula. to be acted upon by a spherically
symmetrical system of forces in addition to its own gravitation. Now it is essential
to the plan of our investigation that we shall be able to make the configuration of the
nebula vary in some continuous manner, and this compels us to retain this generali¬
sation. We shall, however, suppose that when the nebula extends to infinity,
u retains some definite limiting value ux , thus including the free nebula as a
sjiecial case.

* C4. II. Darwin, ‘Phil. Trans.,’ A, vol. 180. p. 1.
t Luc. cit., p. 15.

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

23

§ 22. Let us, in the first place, consider the “series” of nebulae such that u has a
different constant value for each. This series includes a single free nebula, for it
appears from Darwin’s paper that there is a nebula such that u — 1 at every point.
This nebula, it is true, has infinite density at the centre, but this objection disappears
when the innermost shells of gas are replaced by a solid core, the mean density of the
core being equal to three times the density of the gas at its surface, and therefore
finite. Let us, in the first instance, simplify the problem by supposing that the core
is held at rest in space. The boundary equations (35 and 53) which have to be
satisfied at r — B0 now take the forms

(A),,k =0 . . (80),

= 0 .

__ ?*— R0

(81).

independently of the value of n. The value of u in equation (55) being now
independent of r, we may write the solution (56) in the form

. (82),

£ = Ep’'1 + E2W .

in which p, p' are the roots of the quadratic,

t (t — 1) = n (n + 1) — 2u .

We accordingly have

p -f p — 1 j p — p 2 \/ (yi -f- -g-)~ — 2 u j pp — n {ii -j- I) -(- 2 u
Equation (79) now takes the form

Ej (p + n) Bf1*”-1 + E2 (p' + n) R1'‘,+n-1 = 0 ...

while equation (81) becomes

Et (p - n - 1) Bo'1-’1"2 + E2 (p; - n - 1) Bp—2 = 0 . .

The elimination of Ej and E3 from these equations gives

jqy-M' ^ (/ +n)(fl - n - 1)

R0/ — (p + n) (p' - n — 1)

(83) .

(84) .

(85) ,

(80).

(87).

The fraction on the right hand can be simplified by the help of equations (84) ; it
is equal to

2 (u — (n + D3) + (p — &') ( n + i)

2 (u — (n + i)2) — (p - p') {n + D

Now the left-hand member of (87) may be replaced by

cosh {i(p - a') log (Ph/Rq)} + sinh (p - pQ log (RfiRp)}
cosh D(p-p') log (RfiRg)} — sinh (p — p') log

24

MR, J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

so that the equation itself reduces to

tanh {1 - /) log (R,/R„)} = tA ~ « .... (88).

This equation expresses the relation which must exist between RT/R0 and p — p'
(or, what is the same thing, between Rj/Rq and u), in order that p — 0 may be a
solution of the frequency equation.

§ 23. We shall be able to interpret this equation most easily by adopting a
graphical treatment. If we write

x = i (d - /*T> Vi = ~ ~

2 (n + i)

y2 = ^ tanh | y/x log (Ri/R0) K

x + (n + i)2 J ' x/jc
then the equation can be written in the form

V\ = y%-

It will be noticed that y% remains real when x is negative, an equivalent expression
for y2 being

Vz

v' —

- tan { .y/ — x log (Rj/Rq)}.

The roots of equation (87) are now represented by the intersections of the graphs
which are obtained by plotting out y1 and y2 as functions of x. These two graphs

are given in figs. 1 and 2 respectively, the graphs being drawn separately for the
sake of clearness. The graph for yx is, of course, the same for all values of R1(/R0 ;
that for can be varied so as to suit any value of R(/R0 ky supposing it subjected

25

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

to an appropriate uniform extension parallel to the axis of y, and contraction parallel
to the axis of x1 or vice versa. Similarly, different values of (n -fi \) can be
represented by contraction and extension of the first graph.

If we imagine these two graphs superposed, we see that there cannot, under any
circumstances, be an intersection in the region in which x is positive, i.e. (equation
(84)), for a value of u less than (n + |-)3. The lowest value of u for which an inter¬
section can possibly occur is u — 1, and this occurs only when H1/H0 = co . As RL/R0
decreases from infinity downwards, the lowest value of u for which an intersection
occurs will continually increase. Whatever the value of Rj/R0 may be, there are
always an infinite number of intersections in the region in which u > T (n + -f)3.

The values of u found in this way determine the “ points of bifurcation ” on the
linear series obtained by causing u to vary continuously. Thus we have seen that as
u continually increases the first point of bifurcation of order n is reached when u has
a value which is always greater than T (n + T)3. When Rj/R0 is very large, the first
point of bifurcation is of order n — 1 , and its position is given by

«=U . (89).

§ 24. Let us, in future, confine our attention to the case in which R^Rq is very
large. If we gradually remove the restriction that u is to be independent of r, the
various vibrations of frequency p = 0 will vary in a continuous manner. Equation
(55) remains unaltered in form, and, at infinity, it assumes the definite limiting form

e^={»(«+l)-2«,}f . (90).

where u„ is the limit (supposed definite) of u at infinity. It therefore appears that
at infinity the solution for £ approximates asymptotically to that given by equation
(82), if y , y! are now taken to he the roots of

t (t — 1) = n{ii + 1) — 2u„ . . . . . . . (91),

Equation (85) accordingly remains unaltered. Equation (81) takes a form which
is no longer represented by equation (86), but which will impose some definite ratio
upon E1 and E3. It is therefore clear that when Rx is very great, equation (85) can
only he satisfied, at any rate so long as /x and // are real, by taking /x — /x' very
small. Thus a point of bifurcation will again be given by /x — jx = 0, our previous
investigation sufficing to show that this gives a genuine solution to our problem, and
does not correspond to an irrelevant factor introduced in the transformation of our
equations. This point of bifurcation is moreover the first one reached as u increases,
since it is at the point at which /x, y! change from being real to being complex.

VOL. CXCIX. — A.

E

MR. J. II. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

26

We conclude that, independently of the values of u at points inside the nebula,
the smallest value of ux for which a vibration of zero frequency and of order n is
possible is given by

ux = l (n + Y)3 . (92),

or, for all orders, is given by

= H . (93),

the limiting vibration being of order n — 1.

It ought to be noticed that for this limiting vibration equation (82) fails to
represent the solution owing to /x and p! becoming identical. The true solutions for
real, zero, and imaginary values of p. — /x' may be put respectively in the forms

£ = Cj y/r sinh (T (p ~ /x') log eR},

^ — C x\/r log eR,

£ = Ovs/r sin j (/x — p) log eR j ,

in which Cx and e are constants of integration.

At infinity p vanishes to the order of 1/r3, so that dpjdr — — 2 pfr. The value of
g for very great values of r is therefore (equation (50))

g = XT (— 2A + Ra).

At the outer boundary a surface - equation (32) directs us to take B = 0.
Following this out, we find that at infinity A is of the same order as g, and therefore
becomes infinite to the order of y/ r. Suppose, on the other hand, that we start by
taking A = 0, so that B = g XT?’. The value of B now vanishes at infinity to the
order of 1 /y/r, and the surface-equation (32) is satisfied by a motion which vanishes
at infinity. It would therefore appear to be easier to satisfy the boundary conditions
when ?■ is actually infinite than when r is merely very great. This result opens up a
somewhat difficult question, which will be considered in the next section.

Before passing on, we may consider in what way the results which have already
been obtained will be modified, if we suppose the core of the nebula to be free to
move in space, instead of being held fast. For the free nebula ux = 1, so that our
results show that a free nebula will be stable if the core is supposed fixed. The
same must therefore obviously be true when the core is free to move, since a motion
in which nebula and core move as a single rigid body will not influence the potential
energy. When the nebula is not free, fixing the core may be regarded as imposing

MR. J. H. JEANS ON THE STABILITY" OF A SPHERICAL NEBULA.

27

a constraint which does no work ; freedom of the core therefore tends towards
instability. It will be proved in § 28, that a nebula is stable for values of ux which
are less than the critical value, and unstable for values greater than this value.
Assuming this for the moment, we see that a nebula in which the core is free to move
must necessarily be unstable if has a value greater than ljr.

If then, we start with a free nebula and imagine to gradually increase from
ux = 1 upwards, the core being free, it follows that the nebula will first become
unstable when ux reaches some value such that

n > > i

(94).

§ 25. The nebula extending to infinity, let us attempt to find the displacement
which will be caused by a small disturbing potential vn given by

47T

= 9;

in +

j + a r>
1 ]r»+i 1 i

s,

(95).

It is clear that the displacement required will be given by our equations if we
include in V (equation (27)) the terms

47 r

2 n + 1

The equation replacing (42) may be transformed in the manner of § 15, and the
resulting equations will be those of § 16, except that we must replace (52) by

1 cl , .

yh dr ^

= a, —

r=R,

A (p — cr, )

— 1

,-=R,

. . . (96),

and (53) by a similar equation.

If a displacement can be found to satisfy these modified equations, the external
disturbing potential which will be required to hold the system in this displaced
position will be given by equation (94). Now the condition that this displaced
position shall be one of limiting equilibrium is that this disturbing potential must
vanish. To be more precise, vn must be such that the force derived from it vanishes
at every point of the nebula. We must therefore have

a1rn = 0

at all points of the nebula, including r = E,x. Now (95) may be regarded as an
equation giving cq in terms of Tt1. Taking p =crL, as before, we find from (95)

28

ME. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

and this vanishes at all points, including r = Ri; in the case in which Rj is put equal
to infinity, if

(97).

u, -- = o

•=» r

The condition that cc0/rn+1 shall vanish at every point would lead to a similar
equation to be satisfied at the origin, if there were no core. If, however, we retain
the core, it leads to the same equation as was found in § 22 (equation (81), when the
core is held at rest). Thus, our present method of finding a position of limiting
equilibrium has led to a result different from that obtained by the search for a
vibration of zero frequency, in that equation (97) replaces equation (79).

The value of g at infinity is given by equation (82) ; hence we have

r = qo ?

[E,!*-1 + Ej*'-

(9S).

As before, the equation to be satisfied at r = R0 determines the ratio of Ex to E.: :
equation (97) is therefore satisfied if the real parts of p and p' are each less than
unity. Now p, p are the roots of equation (91), hence this condition is satisfied
provided

n {n -f 1) < '2ux . (99).

§ 2G. Let the kinetic and potential energies of a small displacement be given, in
terms of the principal co-ordinates, by

2T = apcp + apcp + . . .

2V = bpc p -f- bc>Xcf + . . .
so that the equations of motion are

ape i — bxx — 0

&c., and _p2 is given by

aYp~ = bl . (100).

The method of §§ 20-24 amounted to finding vibrations such that p2 = 0, and
therefore, by equation (100), solutions of

Zq = 0 . (101).

In § 25, on the other hand, we started with the supposition that the nebula extended
to infinity, so that all the quantities a and b are liable to become infinite. The
equation giving vibrations of frequency p — 0 is no longer equation (101), but is

11, =00 %

and this is obviously more general than equation (101).

(102),

ME, J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

29

It will be noticed that the method of §§ 20-24 is the method which is mathematically
appropriate to the case of a nebula enclosed in a surface maintained at constant
pressure, while the method of § 25 is that appropriate to an infinite nebula. In the
former case, a vibration of frequency p = 0 may represent a real change from stability
to instability ; in the latter case such a vibration leads to an adjacent configuration of
equilibrium, and is, in this sense, a point of bifurcation, but does not denote a change
in the sign of jr.

The General Case of a Nebula extending to Infinity.

§ 27. The method to be followed has been explained in § 18. The general
differential equation is of the sixth order. Four solutions have definite limiting forms
when p — 0 ; the remaining two take singular forms. The former have been examined
in § 16 ; the latter are represented mathematically (p. 18) by functions which do not
approach a definite limit as p approaches a zero value, and physically (p. 11) by
systems of steady currents.

There are six constants of integration, Ei5 E2, E3, E4, E5, E3, of which the two last
belong to the singular solutions. Let us suppose (as is always possible (p. 19)) that
the ratios of these six constants are determined from five of the boundary-equations,
that which is not used being the equation satisfied by £ at the outer boundary. This
remaining boundary-equation now takes the form (cf. equation (56))

Ep/q (Lj) + Eoifq (Itj) fi- E5\f/5 (Itj) -}- Ep f (E,1) = 0 ... (103),

in which the four E’s are definite quantities. The four \f/s must have definite
limiting values (zero and infinity being included as possible values) when 14 = 00.
Thus in equation (103) some terms must preponderate over the others. When the
nebula is isothermal, these terms are the first two. Hence, when the nebula is not
isothermal, it follows from the principle of continuity, that the same two terms must
still preponderate, at any rate for some finite domain including the isothermal
nebula. Otherwise it would be j:>ossible to change the stability or instability of
a nebula by an infinitesimal change in the physical constitution of the nebula.
Hence throughout this domain, equation (103) must reduce to its first two terms,
i.e., must become formally the same as in the case of the isothermal nebula. But
the solution for £(and therefore the functions i/q, \jj.2), remain formally the same in the
general case as in this particular case, and therefore the stability-criterion derived
from equation (103) remains formally the same.

It follows that whether the nebula is isothermal or not (provided always that the
configuration lies within a certain domain of equilibrium configurations) the critical
configurations are given by the two equations (92) and (99).

ME. J. II. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

BO

Exchange of Stabilities.

§ 28. We have now completed an investigation of the configurations at which
a transition from stability to instability can occur, as regards the spherical form, for
vibrations of orders different from zero. It is unnecessary to discuss vibrations of
order n — 0, for the following reason.

Our problem is to determine the changes in the configuration of a nebula which
will take place as the nebula cools, starting from a spherical configuration, supposed
stable. We are not concerned with the succession of spherical configurations, but
only with an investigation of the conditions under which a spherical configuration
becomes a physical impossibility. Now a point of bifurcation of order n = 0 does
not indicate a departure from the spherical configuration. It indicates a choice of
two paths, one stable and the other unstable, and the configurations on both paths
will remain spherically symmetrical.

We have therefore determined already the circumstances under which a transition
from a symmetrical to an unsymmetrical configuration can occur. It remains to show
that there is, in effect, an exchange of stabilities at a point of bifurcation, and to
examine on which side of the point of bifurcation the spherical configuration is stable.

We are going to prove that the spherical configuration is stable for all values of u
less than u0, the lowest value of u at which a point of bifurcation of order different
from zero can occur. Our method will be as follows : Any two equilibrium configura¬
tions can be connected by a continuous linear series of equilibrium configurations, and
u will vary continuously as we move along this series. If one of the two terminal
configurations is stable, and if the linear series can be chosen so that u does not at
any point of it pass through a value for which a vibration of frequency p = 0 is
possible, then we know that the other terminal configuration is also stable.

The value of y, the gravitation constant, has been taken equal to unity. If this
constant is restored, the value of u becomes (equation (54))

u —

277 ypE

dp j dm
dr / dr

Since cr = XT p, we have

dm

dr

= p A (XT) + XT A

dr

For an infinite nebula, the first term on the right-hand of this equation will
vanish at infinity in comparison with the second. Hence we have as the
value of ux

0I _ Lf, 2ir1Pr~

r= go

(104).

ME, j. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

31

If Ave write y — 0 Ave pass to the case of a non -gravitating nebula, and Ave see
that ux — 0 provided the ratio of pr3 to XT remains finite at infinity. Noav Ave can
keep the value of p and XT the same at every point by subjecting the nebula to
an appropriate external field of force, and this field of force will be exactly the same
as the gravitational field which Avas annihilated upon putting y — 0. It is spheri¬
cally symmetrical, and its potential vanishes at infinity to the order of 1/r, so that
it comes within the scope of our previous analysis. For values of y intermediate
between the natural value (y = l) and the value y = 0 we can obtain the same
result by taking a field of force equal to 1 — y times the foregoing. As Ave increase
y from 0 to 1 we obtain a linear series, in which the configuration of the nebula
is unaltered, the nebula being gradually endoAved with the power of gravitation.

For the general configuration of this series, consider the work done in a specified
displacement, Avhich is proportional to S„ at every point. The potential (gravitational
+ that of external field) after displacement Avill be of the form

a + l>y$>,„

Avhere a and b are functions of r and independent of y. The total Avork done against
this field during the displacement is therefore of the form

By,

Avhere B is independent of y and depends solely upon the particular displacement
selected. The work done against the elastic forces is of course independent of y,
and depends solely upon the displacement selected. This work is essentially positive.
The total work is therefore of the form

A + By,

Avhere A is positive and B may (§ 2) be negative. Since y is proportional to uM this
may be written

A -j- B ux . (105).

Suppose this function calculated for all possible displacements. Then we shall find
that for values greater than some definite value of ux it is possible for the Avork
done to become negative. For values of u x less than this critical value, the work
will be positive for all displacements. Hence from the form of expression (105)
it IoIIoavs that the passage of ux through a critical value denotes a real change from
stability to instability, and that the stable configurations are given by the smaller
values of ux.

Recapitulation and Discussion of Results.

§ 29. We haA-e seen that the vibrations of any spherical nebula may be classified
into vibrations of orders n = 0, 1, 2, &c. , a vibration of any order n being such that
the displacement and change in temperature at any point are each proportional to

32

ME. J. H. JEANS ON THE STABILITY OF A SPHEEICAL NEBULA.

some spherical surface harmonic S„ of order n. The frequency of vibration is
independent of the particular spherical harmonic chosen, depending only upon the
order n.

The vibrations of order n — 0 have been seen to be of no importance ; the stability
of the vibrations of orders different from zero has been discussed, in the limiting case
in which the nebula extends to infinity, with the following results : —

Starting from any stable configuration of spherical symmetry, the vibrations of any
order n, different from zero, all remain stable until the function m„, defined by
equation (104), passes through a certain critical value. In any case this critical
value is first attained for a vibration of order n = 1.

For a nebula which actually extends to infinity, the critical value is ux = 1.
When this value is reached we come to a second series of equilibrium configurations,
the form of which will be investigated later. If this value is passed, the configura¬
tion remaining spherical, there will not be vibrations in which the time enters
through a real exponential factor, because the critical vibrations remain of frequency
p — 0, the inertia of the nebula being infinite.

If the radius It 1 of the nebula is regarded as very great but not infinite, this
statement is not true, since the inertia cannot now become infinite. In this case the
first new series of equilibrium configurations is again reached when (n)r=I?i attains a
certain critical value, and the critical vibration is again of order n — 1 . The critical
value of (w),.=El has not been calculated, but when becomes infinite, it has a
limiting value which has been shown to lie between I and 1-g-.

Taking y — 1, we have as the value of ux,

n - L' -pr~

(106).

The question of stability turns entirely upon the value of this function, which may
appropriately be termed the “ stability-function.”

We now see that the whole question of stability depends upon the ratio of the
density to the elasticity at infinity. This result is not hard to understand. In the
first place, since the nebula extends to infinity, we may, so to speak, measure it upon
any linear scale we like. If we measure it on a sufficiently great scale, the nebula
still remains of infinite extent, but the variations in temperature or structure which
occur near the centre can be made to appear as small as we wish, and the solid core
can be made to appear as insignificant as we wish. Thus by measuring any nebula
upon a sufficiently great scale we can make it appear indistinguishable from an
isothermal nebula, and the critical vibration for which 'p = 0 does not disappear from
sight, since in the limit this vibration (measured by £/r) remains finite at infinity.
Further, as Professor Darwin points out, we can make it appear like a nebula in

ME. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

33

which u maintains a constant value throughout.* Passing on, we notice that the
stability function now depends solely upon the ratio of the density to the elasticity.
The different elements of the nebula are attracted towards one another by their
mutual gravitation, and are kept apart by the elasticity of the gas. For certain
values of the ratio of these two systems of forces, it will be possible to find displace¬
ments in which the work done by one system exactly balances that done against the
other, and these are the critical vibrations.

The stability function um is a function only of the quantities determining the
equilibrium configuration of the nebula, and its value may therefore be found from
the equations of equilibrium. We proceed to examine the value.

Evaluation of the Stability Function.
General Case of a Nebula at Rest.

§ 30. We have already quoted Professor Darwin’s result that ux = 1 for an
isothermal nebula at rest, and the considerations put forward in the last section will
probably suggest that the result in the more general case will be found to be
independent of variations in temperature at finite distances, provided only that
the temperature has a definite limit at infinity. We shall, however, examine the
question ah initio , using a slight modification of Darwin’s method, and making the
problem more general by retaining a spherically symmetrical system of external
forces.

We shall denote the potential of this system of forces by V', and use V to denote
the gravitational potential of the nebula itself. The total potential is now V + V',
so that the equation of equilibrium, equation (11), takes the form

yieu)- Jqv + v') = o,

and if M is the mass of the solid core, this can be written

— — (A.Tp) -f- 477 | pr3 dr -J- M — r3- — = 0 . . . . (107).

Differentiating with respect to r,

Write

A

dr

/ r2 d
\ p dr

(XTp) j -j- 47rpr3 —

VTp = e? ,

= 0.

* G. H. Darwin (l.c. ante, p. 16), “ If we view the nebula from a very great distance, . . . the solution
of the problem becomes y = log 2a:2.” Now u = — d2y/dx 2, so that this solution is equivalent
to u = 1. This justifies our statement, and shows at the same time that for any nebula at rest and
in equilibrium um has the critical value ux = 1, provided it is acted upon by no forces except its own
gravitation.

VOL. CXCIX. - A.

F

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

34

and

so that

dr

X ~~ J >• xl>2 ’

_ 1 cl

dr XT r2 do;

then the above equation may be written

d?y

-j-

dor

„ d ( 0 dV'\

tt€j)a — "7- ( r2 -7 = 0 .

dr \ dr ]

(108).

At infinity we are supposing XT to have a definite and finite limit, so that the

L(r^\

dr \ dr !

limiting value of x is l/ATr. Let us further suppose that ~(rz~) has a definite

limit given by

d [ „ dV'

r-

dr\ dr

-AL _ y"

I

109),

and that squares of V" may be neglected. Then the limiting form of (108) at
infinity is

Write

then

and

d2y AiTred
dx* (XLW

V = 7) + log

V"

-3 - 0

X4TU2

2tt

d*y dhj _2 jP , .

dx* ~ dx* 0? 4 ^ °8’ (

dx*

4-71 _ 2er’

(\Tx)i = h?

Equation (1 10) is now transformed into

g + ->-!) + 4 J>g(XT)-^ = 0

d 2

In the special case in which — - log (XT) vanishes, this may be written
d~V 4- — ( n 4- W3 4- W3 4- — — ' ^ = 0

r2 1 7,2 \ y 1 ~y 1 w • • • ‘/>x2t2 * ^ ■

dx 2

and at infinity (be., for very small values of x), the solution is

’ = 2^+ V ieoa(id7 Vg |

(no).

(111).

(112)

where A, B are the two constants of integration.

MR, J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 35

d 2

In the more general case in which — log (XT) cannot be supposed to vanish, it is

clear that this term will vanish at infinity in comparison with the other terms
in (ill), if r] has the limiting value given by (112), and therefore that (112) is the
limit, at infinity, of the solution of (108).

Of the two arbitrary constants, A and B, the former corresponds to the indeter¬
minateness of the linear scale upon which the nebula is measured, the second to the
indeterminateness of the conditions at the inner surface of the nebula. If there is no
core, there is only one value of A/B which will give a finite density of matter at the
centre of the nebula. Further information as to equilibrium configurations can be
found in Professor Darwin’s paper, or in a paper by A. BitterA

For our purpose it is sufficient to know that the second term in y vanishes with x
for all values of A and B. Hence at infinity

V =

V" , , A4TU2

- -I- loo’ —

2\2T2 “ 2tt ’

\3T3A

2ir

^,V"/2A2T2

XT

27rr2

gV"/'2A2T2

?

and hence (equation (107))

u

00

Npl~ -V,,/2A2T2

AT 6

1 +

V"

2\2T2

(113).

Putting V" = 0, we arrive at the anticipated result that the stability function has
a unit value, for every nebula which extends to infinity in such a way that XT
has a finite limit at infinity.

A Slowly Rotating Nebula.

§ 31. The case which is of the greatest physical interest, is that in which the
nebula is not at rest but is rotating in a position of relative equilibrium

Here the arrangement is no longer in spherical shells, so that the foregoing analysis
breaks down. If, however, we suppose the rotation w to be so small that cA may be
neglected, it will be easy to modify the foregoing analysis, so as to take account of
rotation.

We shall still suppose the nebula to extend to infinity, so that we must not suppose
the rotation to be the same at all distances, for in this case a finite value of o> would
imply an infinite velocity of those parts of the nebula which are at infinity. Let us

* ‘ Wied. Ann./vol. 16, p. 166.
F 2

36

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

suppose that at infinity the linear velocity approximates to a finite limit, so that we
may write

o) = 91 jr

for all values of r greater than a certain amount.*

So long as we are only concerned with configurations of equilibrium and vibrations
of frequency p — 0, the rotation may be allowed for by the introduction of a force of
amount orr sin 6 per unit mass, acting perpendicular to the axis of rotation ; or, what
comes to the same thing, by the introduction of a potential

| (1 — P2) forWr,
or

(1-P2) V'

where, for all values of r greater than a certain value,

V' = f H2 log r . (114).

Let us examine separately the two effects arising from the two terms of this
potential, beginning with the term — P2V'. There will in this case be a correction
to be applied to all equations, and this correction will consist of the addition of
a small term containing orP3. Let us suppose that all symbols which have so far
denoted functions of r, denote in future the mean value of the corresponding
quantities averaged over a sphere of radius r. For instance, p is no longer the
density at distance r from the centre, but is the mean density over the sphere of
radius r. The density at any point will be of the form p + urP ,2p.:, where p, is
a function of r. We may in every case equate the coefficients of different harmonics,
and by equating the coefficients of terms which do not contain the terms &rP2, we
shall obtain the same equations as were obtained in the case of a> = 0, except that
the meaning of every term is altered.

The equations derived from the parts which do not contain w will suffice, as before,
to determine p, so that the values of p are of the same form as before, except that
the quantities involved have a slightly different meaning. Hence the stability
criterion is still given by the value of the stability function ux ; while equation (107)

* This particular law is chosen for examination because it leads most quickly to the required result.
The case in which w vanishes at infinity more rapidly than 1/r is covered by taking 9. = 0. Here,
however, the angular momentum vanishes in comparison with the mass, and it is not surprising to find
that a rotation of this kind does not affect the question of stabitity. The case in which w vanishes less
rapidly than 1 jr is physically impossible, since it gives an infinite linear velocity at infinity, but may be
theoretically included in the case of 9 = oo .

Any special assumption about the value of w at infinity must, however, disappear when we turn to the
case of a finite nebula (§ 26), in which may be appropriately supposed to correspond to the surface
velocity wR,.

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

37

remains true, if the new meaning is given to the symbols in each case. We conclude
that the question of stability is not affected by the potential — PaV'.

The remaining potential term is the spherically symmetrical term V'. The total
potential may now be taken to be V + V', and this potential, besides being spherically
symmetrical, satisfies the condition which was postulated in the determination of the
criterion of stability ; namely, that its radial differential coefficient shall vanish at
infinity to the order of 1/r. The value of the derived function V'7 (equation (109)) is

V" = Li y- (r2 -y- ) — -In3, by equation (] 14).

Hence the stability function is given by (cf. equation (113))

, H2

V zz: 1 4- -

' ^ 3\2T2 ’

WTe have therefore found that when an infinite nebula is rotating, with such
angular velocities that the linear velocities at infinity have the limiting value H, the
value of Mj, is greater than unity no matter how small H may be. This result has
only been obtained on the supposition that ofi may be neglected. We have obtained
no information as to what happens when ofi is taken into account, i.e., when the
square of the “ ellipticity ” of the nebula is taken into account.

Influence of Viscosity.

§ 32. No account has so far been taken of the viscosity of the gas. The terms
arising from viscosity which may be supposed to occur in the true equations of
motion, will contain the coefficient of viscosity (y), and will in each case depend on
velocities and not on displacements. Hence viscosity enters the equations of motion
through the factor yip. The vibrations for which p = 0 are accordingly unaffected
by viscosity, and since it is upon the existence of such vibrations that the whole
question of stability turns, it is clear that the results already obtained must remain
true even in the presence of viscosity.

It can be shown that equations (24) to (26) specify a principal vibration, whether
the gas is viscous or not. The result is stated without proof, as the proof is rather
lengthy, and has no bearing upon the main question under discussion.

A Nebula in Process of Cooling.

§ 33. In the mathematical investigation we have been concerned with vibrations
about a position of absolute equilibrium. In nature, no such position of absolute
equilibrium will occur ; the condition of the nebula will be incessantly changing.

38

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

Let us suppose the temperature of the nebula to be continually cooling, owing
either to radiation of heat from its surface or to a process of quasi-evaporation such
as is described in Professor Darwin’s paper (§13 or p. 66). Since the gas (or quasi¬
gas) is not a perfect conductor, the nebula will not at any time be in perfect thermal
equilibrium. The changes in density of all parts, and in the temperature of the
inner parts of the nebula will, so to speak, lag behind their equilibrium values as
determined by the changes in the temperature of the outer part of the nebula. It
is, therefore, clear that so long as the nebula is cooling, the ratio of the density to
elasticity in the outermost layers of gas will be greater than that calculated upon
the assumption of perfect equilibrium. This “lag” accordingly decreases the value
of the stability-function, and so supplies a factor which tends to instability.

Summary and Discussion of Pvesults.

§ 34. Let us now examine to what extent we have found solutions of the two
problems propounded in § 4.

Firstly, as regards the stability of a spherical nebula of very great size, of which
the outer surface is maintained at constant pressure. We have found that the
stability-function for such a nebula (in the limiting case in which the outer radius is
infinite) has a unit value when the nebula is in equilibrium and at rest. This value
is increased by allowing for the “ lag’ in temperature caused by the cooling of the
nebula. It is also increased by a rotation of the nebula, at any rate so long as this
rotation is small. The nebula will become unstable as soon as the stability-function
becomes greater than a certain value, which has not been calculated, but is known to
be between 1 and 1|-. The investigation of § 23 leads us to expect that the critical
value of the stability -function will increase as Px decreases, although this has only
been strictly proved for a single case.

It is therefore possible that, even when the nebula is non-rotating, the temperature-
lag may be sufficient to make the nebula unstable. If sve disregard the temperature-
lag, it seems probable that a small rotation will suffice to bring about instability.
This latter question, however, deserves more detailed examination.

§ 35. Let us suppose that the nebula starts from rest in a configuration of absolute
equilibrium, and that the rotation is gradually increased. In this way we obtain a
linear series of configurations of relative equilibrium. When the rotation is small,
the configuration, instead of being strictly sjiherical is slightly spheroidal. The
series we are considering is therefore the analogue of the series of Maclaurin
spheroids of an incompressible fluid. So long as the rotation remains small, we may
separate the two terms of the rotation-potential in the manner explained in § 31.
We may, in fact, suppose our analysis still to apply as if the configuration remained
spherical, and the only effect of the rotation is to increase the value of the stability-
function. For larger values of or, all our results are subject to a correction of the

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

39

order of aA For small values of or, the value of or1 will be proportional as we have
seen (§ 31) to ux — 1. so that this correction may be supposed to be proportional to
iuw — l)2. The first points of bifurcation of orders 1, 2 occur (in the spherical
configuration) at — 1 = 9, 2^ respectively, where 6 is known to be less than
Now it would seem to be fairly safe to neglect 02, but even if we waive this point, it
will be admitted that the correction of the order of (w« — I )s cannot be so great as to
change the order in which these two points of bifurcation will occur.

We therefore see that a rotating nebula will become unstable for a comparatively
small value of ad, the critical vibration being of order n = 1. The new linear series
is one in which (except for the spheroidal deformation caused by the rotation) the
surfaces of equal density remain spheres, which are no longer concentric. The linear
series of order n — 2 will accordingly be unstable : this is the analogue to the series
of Jacobian ellipsoids in the incompressible fluid.

§ 36. The case of a nebula which actually extends to infinity is much simpler.
Here the value of ux is again unity, and this value is increased, as before, either by
temperature-lag or rotation. Every point at which ux is greater than unity is in one
sense a point of bifurcation, since starting from this point there is a series of
unsymmetrical equilibrium configurations. Strictly speaking, these points do not
indicate an exchange of stabilities, for the critical vibrations remain of frequency
p = 0 even after passing the point. They possess, however, the property that a
critical vibration, if once started, will continue increasing, since the forces of
restitution (of whichever sign) vanish in comparison with the momentum of the
vibration.

§ 37. Let us now try and examine which of these two hypotheses is best capable of
representing the “ primitive nebula ” of astronomy. Imagine a sphere S drawn in
the nebula, the radius being a , and the pressure at this surface n. The matter
inside S is to form a spherical nebula of finite extent, bounded by a sphere over
which the pressure is tt, and this matter is to be of a density sufficient to warrant us
in assuming the gas-equations at every point. The surface S will be continually
traversed by matter, but this will be of no consequence if the losses and gains
balance in every respect. The matter outside S must supply the pressure tt, and
will also, as was explained in the introduction (§ 3), influence the matter inside S by
its motion.

Imagine the matter inside S to be executing a small vibration, and consider two
extreme hypotheses as to the behaviour of the matter outside S.

Suppose, in the first place, that the matter outside S is such that it and the
matter inside S together form a perfect spherical nebula at rest. Then the motion
of the matter outside S is given by the equations of vibration of such a nebula, and
the influence of this matter upon that inside S is exactly that required in order to
enable the matter inside S to execute the vibrations given by the equations of an
infinite nebula.

40

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

Suppose, next, that the matter outside S consists mainly of molecules or of masses
of matter which are describing hyperbolic or parabolic orbits, or which come from
infinity and after rebounding from the nebula return to infinity. Suppose, further,
that the interval during which such a mass is appreciably under the influence of the
matter inside S is so small that it is not appreciably affected by the motion of the
latter. In this case the matter outside S may he regarded as arranged at random,
independently of the vibrations of the matter inside S ; it will not, as under our first
supposition, take up the motion of the matter inside S to any appreciable extent.
Hence the matter outside S will exert no force upon that inside S except the
constant pressure v, and the vibrations of the matter inside S wall be those of a
spherical nebula of finite size, bounded by a surface at constant pressure n.

These two extreme hypotheses lead, as wTe can now see, to the two conceptions of a
nebula put forward in § 4. In nature the truth will lie somewhere between these
two hypotheses, and it is by no means easy to decide which of the two gives the
better representation of an actual nebula. We shall, however, be within the limits
of safety if we assert of an actual nebula only those propositions which are true of
both our ideal nebulae.

§ 38. We may accordingly sum up as follows : —

(i.) A nebula at rest and in absolute equilibrium in a spherical configuration will
always be stable.

(ii.) Such a nebula may become unstable as soon as the temperature-lag is taken
into account.

(iii.) There will be a linear series of configurations of relative equilibrium of a
rotating nebula, starting from a non-rotating spherical nebula (supposed
stable), and such that the configuration is symmetrical about the axis of
rotation. This linear series corresponds to the series of Maclaurin
spheroids.

(iv.) The first point of bifurcation on this series occurs for a comparatively small
value of the angular rotation.

(v.) The second series through this point is one in which the configurations
possess only two planes of symmetry. Initially the configuration is such
that the equations to the surfaces of equal density contain only terms in
the first harmonic in addition to those required by the angular rotation.

(vi.) There is a linear series which corresponds to the series of Jacobian ellipsoids,
each configuration possessing three planes of symmetry. The point of
bifurcation at w^hich this series meets the series mentioned in (iii.) is a point
at which the angular rotation is much larger than that at the point of
bifurcation mentioned in (iv.).

(vii.) This latter linear series appears to be always unstable.

ME. J. H. JEANS ON THE STABILITY OF A SPHEKICAL NEBULA.

41

The Unsymmetrical Configurations of a Nebula.

The Second Series of Equilibrium Configurations.

§ 39. Let us now try to examine the second series of equilibrium configurations,
which, as we have seen, is a series of stable configurations replacing the series of
Jacobian ellipsoids. In this way we shall be able to gather some evidence with a
view to forming a judgment whether the behaviour of the nebula after leaving the
symmetrical configuration is such as is required by the nebular hypothesis.

Let us suppose, in the first instance, that the symmetrical configuration from which
this series starts is one in which there is no rotation, so that the configuration is one
of perfect spherical symmetry. If the nebula is one in which cooling takes place
very slowly, the configuration of the nebula will always be very approximately an
equilibrium configuration. This configuration will be one of the spherically
symmetrical series until the first point of bifurcation is reached ; after this the
configuration will change so as to move along the other series, which passes through
this point.

Now we have already found the manner in which the configuration first diverges
from spherical symmetry : in other words, we have a knowledge of the unsymmetrical
series in the immediate neighbourhood of the point of bifurcation. If then, we can,
by some method of continued approximation, obtain a more extended knowledge of
this series of configurations, we shall be able to trace the motion of a nebula which
is cooling with infinite slowness, and in this way form some idea of the motion to be
expected in the more general case.

Let us assume, as a general form for the “ series ” now under discussion,

P — Po + Pl^l + p-p 2 + P3P3 + . . •

where Ps is the zonal harmonic of order s, and p0, pL, p.2 are functions of r and of
some parameter a. This parameter determines the position of any particular
configuration in the series. We shall suppose that at the point of bifurcation a = 0,
and we then know that when a is very small the limiting form of p is

P — Po +

In the notation which lias been in use throughout the paper, we find that
corresponding to the density distribution given by equation (72) the gravitational
potential at the point r , 9 is

where

V — $0 "U 6*1-Pl ~b 02^2 + ^3^3 ~b • •

47T

(116),

9,. =

2s + 1 l_r'!+1JR

1 psr“+z dr -f r’^ Ps'^

11— l

VOL. CXC1X. — A.

G

(117).

42

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

The functions px, p.3 . . . . are to be determined from the condition that V and p
shall satisfy the three equations of equilibrium, which are of the form

da _ d\

p dx dx

An Isothermal Nebula.

§ 40. Let us suppose, for the sake of simplicity, that the nebula is at uniform
temperature, and extends from r = 0 to r = oo . We have already seen (equation (77))
that the critical vibration for a nebula initially isothermal, is one in which the nebula
remains isothermal. Hence it follows that ifi a nebula changes its configuration
through coming to a point of bifurcation, when moving on a series of isothermal
and spherical configurations, then the new series will also be one in which the
equilibrium is isothermal.

We may now write ct = «p, where k is a constant, and the three equations of
equilibrium become equivalent to the single equation,

or

k log p — N + c ,

V+c

p = e K

(118).

Now the series in question is, as we have seen, approximately represented, near to
the point of bifurcation, by taking only two terms of (115), and consequently only
two terms of (116). In this case equation (118) becomes :

Po + PiJ'i — e j1 + jL + i („’ Pi) +

. . (119).

Equating coefficients, we find that p0 is given by the equation

0,1+ r

Po = e ' « ,

the same equation as in the case of' perfect spherical symmetry. Also px is given by
the equation

Pi - Ji¬
lt will be easily verified that this equation is exactly equivalent to our former
equation (38). The equation contains an arbitrary multiplier in its solution. This
may be taken to be a, the parameter of the series, so that we may write

Pi = «oq,

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

43

where oq is a completely determined function of r. Thus, as far as a , the solution is
seen to be

P = Po + «0TPi-

We shall now show that, as far as a3, the solution is

p = po -fi «2cr 02 ~h GoqPi -T croqP 2 . (120).

The substitution of this in equation (118) leads to

Po + + «o'1P1 + cr<r2P2

= e'V ( 1 + f5b Pj + 1 Pf> p,y + . . . + -*?i P2 +

fC

K

K,

■ + . ..j,

K J

where cf>l stands in the same relation to oq as does 6X to P\- The right-hand member
of this equation is equal to

eYL +*;£ + *- + *.Pl + (f* + Jf£)P. +

' L K- K K \ K K~J J

in which the unwritten terms are of degree at least equal to 3 in a.

Neglecting a3 the equation is satisfied if

_ f 1 0r , 003

Po ~ e K » °"o2 — do j i ^2 + —

cr,

Po0i

_ Po02 I 1 Po0i~

cr.i — - ~r 3 "g-

(121).

(122),

(12.3).

These equations determine o-02 and oq uniquely.

It is obvious that this method is capable of indefinite extension, and that the
general form of configuration in the series will be given by

P = do + a2(To2 + a4cr04 + • • • + (a<Xi + &3cr13 + oUoq- -f . . .) P2

-j- (ore To + cdcr24 + . . . ) P2 + (aJo-o + a5cr3S -j- . . . ) P3 -fi &c. . (124).

§ 41. Let us examine in greater detail the solution as far as a3, this being given by
equation (115). The important question, as will be seen later, is the determination
of the sign of cr2. We therefore pass at once to the consideration of equation (123).
Written out in full, this becomes

cr, =

47rpn fir ,

OK 1

G 2

44

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

This equation may be transformed in the same way as equation (39). If we
write

«'r i x Pn<f>l\

y = ZVr'-*~*)'

we find that the above equation is equivalent to (cf equations (47), (48), (49),
(54), (56))

dry Gy

~d? ~~7~~

477 our

together with the two equations

'1 *a\'

r~ dr ^ ^

r6 Jr (; yr s)

4? { y - 1 } • ' '

■ • (125),

= 0 .

0

. . (126),

— 0 .

. . (127).

,■ = 0

Writ

tmg

equation (119) becomes

y _ fW'

d

dr

■y y

h (6 - 2 u) =

_ ,?’Pn0r

(128).

Referring to the table of values for u, which will be found on p. 15 of
Professor Darwin’s paper, it appears that u increases from a zero value at the origin
up to a maximum value of about 1'66 ; it then decreases to a minimum of about "8,
and after this increases to 1, its value at infinity. Thus the factor 6 — 2 u has a range
of values from 6 to about 2§.

Now the solution of

&y

dr~

4 n(n + 1) =

is easily found to be
477/

y —

- 4til J _L [' Pah:r'u+ a cjrf + rn f

2n + 1 [r,i+1 Jo 3/d ^

OKT7

3/e2

(129)

^dr' | + Cp-"" + C3r’'+1 .

(130),

in which Cj and 03 are constants of integration, which may at once be put equal to
zero, if n is positive, and if y is to satisfy conditions (126) and (127).

Comparing (128) with (129), we see that if u had a constant value v0 at every
point of the nebula, the value of y would be given by equation (130), in which C1? Ch
would lie put equal to zero, and n would be the positive root of

n (n -f- 1) = 6 — 2w0,
provided only that 6 — 2w0 were positive.

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

4,r)

For the range of values for u0 from n0 = 0 to w0 = F66, the value of n would have
a range of values from 2 to 1*2. Thus the form of solution is materially the same for
all of these values of u0. It will be seen without difficulty that the solution of (128),
in which u has not a constant value, but varies over the range from 0 to 1 • 6 6 as r
varies, will be such that the graph expressing y as a function of r will present the
same features as are common to the graphs given by equation (130) for ranges of n
from 2 to 1*2.

Now the value of y given by equation (130) is positive for all values of r, hence we
infer that the solution of (128) is such that y is positive for all values of r. We
therefore have, for all values of r,

cr3 = 1 ~ 3 + a positive quantity,

so that <r3 is positive for all values of r.

§ 42. We therefore see that the initial motion, in which u and A are each
proportional to the first harmonic, will first break down owing to the introduction of
terms Involving the second harmonic. The sign of these terms is such that there is,
in all the shells of which the nebula is composed, a diminution of density in the
equatorial regions, and a condensation at both poles, which must be added to that
given by the terms involving the first harmonic.

The nature of this motion will become clearer upon a reference to fig. 3. This
figure consists of the four curves*

r = cIq r = a0 + a1P1

r = a0 -f- auP l fi- u3P3 r — a0 -f- cdnP 1 fi- a 3P3,

and these may be supposed to represent curves of equal density in the three stages.
It is easy to see that of the pear-shaped surfaces of equal density, the equations of
which contain the two first harmonics, some will be turned in one direction, and some
in the other. For if they were all turned in the same direction the centre of gravity
could no longer remain at the centre of co-ordinates. Thus, if the narrow ends of
these pear-shaped figures point in one direction at infinity, we must, as we go
inwards, come to a place at which they have the transition shape, namely, ellipsoids
of revolution, and after this they will point in the opposite direction.

It appears, therefore, that the initial motion is such as to suggest the ultimate
division of the nebula into two parts, this division being effected by the outer layers
condensing about one radius of the nebula, so as to leave room for the ejection of a

* The particular values for which the curves are drawn are in the ratio a0 = 11, a\ — 2, «xi = 5, a2 = 2,
a\ i = 7, a'. 2 = 4. Thus the equation of the last curves are in polar co-ordinates,

r = (10 + 5 cos 9 + 3 eos2 6), r = — (9 + 7 cos 6 + 6 cos2 6).

46

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

central nucleus in the direction of the opposite radius. Whether or not actual
separation takes place would probably depend on the amount of the angular velocity.

It is of interest to compare the result just arrived at, with the corresponding
result found by Poincare for the motion when an ellipsoid of Jacobi first becomes
unstable. # This is described as follows : —

“ La plus grande portion de la matiere semble se rapprocher de la forme spherique,
tandis cpie la plus petite portion de cette merne matiere sort de l’ellipsoide par
l’extremite du grand axe, comme si elle voulait se separer de la masse principale.”

Thus, although the initial motions are, since they start from different configurations,
necessarily different, yet it would seem as if the final result was very much the same
in the two cases. In either case we have a diminution of matter in the equatorial
regions, suggesting the ultimate division of the mass into two, and in each case these

two masses are of unequal size, a result which could hardly have been foreseen
without analysis.

§ 43. If the rate of cooling of a nebula is appreciable, the motion will not be along
a “series” of equilibrium configurations. The value of p, the frequency which is
nearest to instability, will be changing at a finite rate, and may run to some distance
beyond the zero value, before the deviation of the nebula from the spherical shape is
sufficient to invalidate the analysis of our paper. In this case we can imagine the
first unstable vibration, that for which p — 0? being overtaken by other unstable
vibrations of greater and greater frequency, the corresponding velocity of divergence

* ‘Acta Mathematica,’ vol. 7, p. 347.

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

47

from spherical symmetry becoming continually greater. It is therefore quite
conceivable that the motion may become adiabatic at an early stage, and it is possible
that it may be better imagined as a collapse or explosion, rather than as a gradual
slipping from a spherical state of equilibrium into and through a series of
unsynnnetrical states of equilibrium.

But an examination of the physical character of the motion will show that in this
extreme case, as also in any intermediate case, the motion must be, in its essentials,
the same as that which has been found for the other extreme case, namely, that of
infinitely slow cooling and perfect thermal equilibrium. In the spherical state, the
outermost layers of gas may be regarded as stretched out in opposition to their
gravitational attractions, being maintained in this state by the elasticity of the gas.
The balance between these two agencies (which is, speaking loosely, measured by the
stability function, ux) must be supposed to be continually changing, and instability
always results from the same cause, namely, that the elasticity of these outer layers
becomes inadequate to resist the gravitational tendency to collapse. In every case
the outer layers concentrate about a single radius of the nebula, the axis of harmonics
(6 = 0 in equation (72)) and so increase the pressure along this radius, while
decreasing that along the opposite radius (6 = tt). This pressure acting upon the
inner layers of gas and the core sets them in motion, and in this way we have the
tendency to separation into two nebulae.

A Nebula in “ Isothermal -adiabatic ” Equilibrium.

§ 44. A nebula which consists of an isothermal nucleus with a layer in convective
equilibrium above it, is said to be in “ isothermal-adiabatic ” equilibrium. At the
surface at which the law changes from the adiabatic to the isothermal, the quantities
<77. T and p must all be continuous.

The isothermal part is capable of executing a vibration of frequency p = 0 while
remaining in isothermal equilibrium throughout, provided the forces acting upon it
from the adiabatic part are the same as would act if the adiabatic part were replaced
by an isothermal part in such a way that the whole made up an infinite isothermal
nebula. If the nebula is rotating, the amplitude of vibration of the infinite nebula
will vanish at infinity proportionally to some inverse power of r, this power increasing
with the rotation. For sufficiently large rotations, the vibrations may be regarded
as inappreciable except over the original isothermal nucleus, so that the vibration is
approximately unaltered when the outer layers are again replaced by layers in
convective equilibrium.

We see, therefore, that an “ isothermal-adiabatic ” nebula may become unstable, for

sufficiently large rotations, through a vibration of order n— I. No attempt is made

to obtain any numerical results. We can, however, follow up the subsequent motion

in the same way as in the case of an isothermal nebula.

«/

48

ME. J. H. JEANS ON THE STABILITY OF A SPHEEICAL NEBULA.

Over the part of the nebula which is in adiabatic equilibrium, the relation between
density and pressure is

~ = cpy,

where c is a constant, so that the equations of equilibrium become

7_o dp dV
CVP lx = * ’ &c-’

and are therefore equivalent to the single equation

r' = v - v0,

7-1

where V0 is the potential of the outer boundary of the nebula. This takes the form

\r \t _ cr/Po/ \ -i i / i \ Fi p i (y ~~ i)(ry — -) (p\ p r i

+ (y - 1) Pj P. + ■ ■

Pij
or,

- v» + + ■ ■ ■ = { 1 + (7~T~- (£

It is obvious that equation (124) again gives the general form of solution, and that,
as far as cr, the equations are (cf equations 121, 123)

e« - v0 = JZl . (131),

<h = cypj-' (^) (132),

* = \2 + i - J [fj] . (133),

_ cy (y - 2) ! [a\ Y , 0 .

~ 6 . (lo4^-

Writing k for ey/V_\ we see that equations (132) and (133) may be written

o-i =

Po$i

Pu^i | 1 /.) \

^2 — „ + f (2 — y)

O

(135).

(133).

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

49

These are equations similar to (122) and (123); the last term in (136) is different
from the last term in (123), but both terms agree in being invariably positive. Hence
it appears that the question of the sign of oq turns, as in § 37, upon the sign of the
factor (6 — 2 u). We can no longer actually evaluate this factor, as in § 37, but it
seems to be safe to infer from analogy that it will be positive at every point, and this
in turn shows that oq must be positive at every point. Hence it appears probable
that the motion will be that described in § 38.

Rotating Nebula.

§ 45. The equations of an unsymmetrical series starting from a symmetrical
configuration in which there is a finite amount of rotation would be extremely
complicated, and no attempt to handle them is made in this paper. The correction
for a small rotation will clearly consist merely of an increase in the terms containing
the second harmonic, so that the general shape of the curves will be similar to that
of the last two curves of fig. 3.

Little difficulty will be experienced in imagining the shape of curves appropriate
to larger rotations.

Problems of Cosmic Evolution.

Infinite Space filled with Matter.

§ 46. A limiting solution of the equations of equilibrium (corresponding to A = co ,
B = co in equation (114)) gives a nebula in which the density is constant every¬
where. This solution may be supposed to represent infinite space filled with matter
distributed at random. If space has no boundary there is presumably no need to
satisfy a boundary-equation at infinity, so that p may have any value ; if, however,
this equation must be satisfied the only solution is p = 0.

Let us consider the former case. Space is filled with a medium of mean density p
and of mean temperature T. Since the space under consideration is infinite, we may
measure linear distances on any scale we please, and, by taking this scale sufficiently
great, we can cause all irregularities in density and temperature to disappear. We
may, therefore, suppose at once that the density and temperature have the constant
values p and T.

The equations of motion for small displacements referred to rectangular axes are,
in the old notation (cfi. § 6), since V0 and are constants,

(137),

or, operating with d/dx, d/dy, d/dz, and adding

VOL. CXCIX. — A. H

(fifi _ _ i/\' dm'

dt 2 dx p0 dx ’

50

MR. J. H. .JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

d2A
dt 2

= V-V'

Po

(138).

Since V ' is the gravitational potential of a distribution of density — Ap ( cf. § 6), we
have

V3V' = 4,it AP,

(139),

while if we suppose, for the sake of simplicity, that the motion is adiabatic, so that
the ratio of pressure to density changes at a constant rate k, we have (cf equation
(3), p. 5)

y v = /cvy = - kPov*a.

Hence equation (138) becomes

d~A

— -7 — 4:irpA — kVA = 0 . (140).

The simplest solution of this is of the form

where

A =

1

gi (p>t ± qr)

0

T

p2 + 4:17 p

K

(141) ,

(142) ,

and the general solution can be built up by superposition of such solutions.

Now solution (141) gives A = 0 at infinity, provided q is real, and therefore
provided qr + ^np is positive, a condition which admits of p being imaginary.
There is therefore a possible motion, which consists of a concentration of matter
about some point, the amount of this concentration vanishing at infinity, and the
amount at any point increasing, in the initial stages, exponentially with the time.

We conclude, therefore, that a uniform distribution in space will be unstable,
independently of the mean temperature or density of this distribution. #

The Evolution of Nebulce.

§ 47. We can also see that a distribution of matter which is symmetrical about a
single point will be equally unstable. For, if this distribution of matter were perfectly

* An interesting field of speculation is opened by regarding the stars themselves as molecules of a
quasi-gas. If space were Euclidean and unbounded, there would be no objection to this procedure, and
we should be led to the conclusion that the matter of the universe must become more and more concen¬
trated in the course of time. If space is non-Euclidean, this concentration might reach a limit as soon as
the coarsegrainedness of the structure attained a value so great that the distance between individual
units became comparable with' the radii of curvature of space. In any case, it may reach a limit as soon
as an appreciable fraction of the space in question becomes occupied by matter.

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

51

homogeneous, the whole mass of matter would form a spherical nebula of literally
infinite extent, and would therefore be in neutral equilibrium. The introduction of
even the smallest irregularities into this structure is equivalent to the application oi
an external field of force. This, as has already been seen, will destroy the spherical
symmetry, and it can easily be seen that the motion from spherical symmetry is such
as to lead to a concentration of matter about points of maximum density.

It appears, therefore, that the configuration which will naturally he assumed by an
infinite mass of matter in the gaseous or meteoritic state consists of a number of
nebulae (i. e. , clusters round points of maximum density). We may either suppose
the outer regions of these nebulae to overlap, each nebula satisfying the gas -equations
by being of infinite extent, or we may suppose the nebulae to be distinct and of finite
size, the interstices being filled by meteorites or other matter, which by continual
bombardment upon the surfaces of the nebulae supply the pressure which is required
at these surfaces by the equations of equilibrium.

§ 48. What, we may inquire, will determine the linear scale upon which these
nebulae are formed ? Three quantities only can be concerned : y the gravitational
constant, p the mean density, and XT the mean elasticity. Now these quantities
can combine in only one way so as to form a length, namely, through the expression

XT

VP

>

of which the dimensions will be readily verified to be unity in length, and zero in
mass and time. We conclude, then, that the distance between adjacent nebulae will
be comparable with the above expression.

Now the value of y is 65 X 10-9, and if we assume the primitive temperature
to be comparable with 1000° (absolute) we may take XT = 109 (corresponding
accurately to an absolute temperature of 350° for air, 2800° for hydrogen). If we
take the sun’s diameter as a temporary unit of length, the earth’s orbit is (roughly)
of diameter 200. If we suppose the fixed stars to be at an average parallactic
distance of 0'5" apart, measured with respect to the earth’s orbit, we find for their
mean distance apart, about 4 X 107 sun’s radii. The density of the sun being,
in C.G.S. units, roughly equal to unity, we may, to the best of our knowledge,
suppose the mean density of the primitive distribution of matter to be about
(4 X 107)~3, or say 10~23. Substituting these values for y, XT and p, we find as the
scale of length a quantity of the order ot 1019’5 centims. The distance which
corresponds to a parallax of CB5" would be about I018'6 centims. It will therefore be
seen that we are dealing with distances which are of the astronomical order of
magnitude.

MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

5

9

The Evolution o f Planetary Systems.

§ 49. Let us now regard a single centre, together with the matter collected round
it, as the spherical nebula which is the subject of discussion. On account of the
way in which it has been formed, this nebula will, in general, be endowed with
a certain amount of angular momentum. We have seen that a primitive nebula of
this kind may be supposed, under certain conditions, to become unstable. We have
also seen that the motion, when the nebula becomes unstable, is such as to strongly
suggest the ejection of a satellite.

As a nebula cools the rotation increases, owing to the contraction of the nebula,
and fi also increases. Thus the quantity H2/3X2T2, which measures the rotational
tendency to instability, has a double cause of increase ; firstly owing to the increase
in H, and secondly owing to the decrease in T. We can accordingly imagine the
primitive nebula becoming unstable time after time, throwing off a satellite each time.

In the usually accepted form of the nebular hypothesis, the rotation is supposed to
be the sole cause of instability, so that the system resulting from a single nebula
ought theoretically to be entirely symmetrical about an axis. On the view of the
present paper, there is no reason for expecting this symmetry. For large rotations
of the primitive nebula, the configuration of the resultant planetary system will
approximate to perfect symmetry, but for small rotations, a slight irregularity
occurring at the. critical moment, at a point out of the equatorial plane, may produce
a satellite of which the orbit is far removed from the equatorial plane.

In conclusion, two particular cases of “ irregularities” may he referred to. If the
nebula is penetrated by a wandering meteorite, at a moment at which it is close to a
state of instability, the presence of the meteorite will constitute an irregularity, and
may easily result in the formation of a satellite. And if a quasi-tide is raised in the
nebula by the presence of a distant mass, the same result may be produced. In the
former case, the plane of the satellite would, if the rotation is sufficiently small,
be largely determined by the path of the meteorite ; in the second case, by the
position (or path) of the attracting mass. It would not, in either case, depend much
upon the axis of rotation of the nebula.

Conclusion.

§ 50. To sum up, it appears that the behaviour of a gaseous nebula differs in
at least two important respects from that of an incompressible liquid. In the first
place, it differs as regards the amount of rotation which is required to produce
instability, and, in the second place, it differs as regards the disposition of the orbits
of the planets which will be formed out of the primitive nebula. It will be noticed
that no definite numerical results have been obtained ; my aim has been to obtain
qualitative rather than quantitative results, so as to show, if possible, that the

ME. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA.

53

results to be expected for a gaseous nebula are of so much more general a kind than
those usually inferred from the analogy of a liquid mass, that no difficulty need be
experienced in referring existent planetary systems to a nebular or meteoritic origin,
on the ground that the configurations of these systems are not such as could have
originated out of a rotating mass of liquid.

In conclusion, I wish to express my indebtedness to Professor Darwin for much
assistance which I have received from him throughout the course of my work.

INDEX SLIP.

Callejjdab, Hugh L. — Continuous Electrical Calorimetry.

Phil. Trans., A, vol. 199, 1902, pp. 55-1 IS.

Calorimetry — Continuous Electrical Method.

Calieitdab, Hugh L. Phil. Trans., A, vol. 199, 1902, pp. 55-148.

Conductivity of Liquid — Electrical Method of Measurement.

Calxendae, Hugh L. Phil. Trans., A, vol. 199, 1902, pp. 55-148.

Current— Absolute Measurement by Electrodynamometer.

Called dab, Hugh L. Phil. Trans., A, vol. 199, 1902, pp. 55-143.

Specific Heat of Water — Variation determined by Electrical Method.

Caixendar, Hugh L. Phil. Trans., A, vol. 199, 1902, pp. 55-143,

Thermometry — Compensated Box for Electrical Thermometry.

Caxlexdab. Hugh L. Phil. Trans , A, vol. 199, 1902, pp. 55-143.

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[ 55 ]

II. Continuous Electrical Calorimetry.

By Hugh L. C allend ar, F.R.S., Quain Professor of Physics at University College ,

London

Received November 18, 1901, — Read February 6, 1902.

Table of Contents.

Part I. — Introduction.

Page

(1.) General account of the Origin and Progress of the Investigation . 57

Part II. — Electrical Measurements.

A. Potential.

(2.) Advantages of the Potentiometer Method . 60

(3.) Description of the Potentiometer . 63

(4.) Method of Testing . 65

(5.) Method of Calibration . 66

B. Resistance.

(6.) The Lorenz Apparatus . 71

(7.) Values of the Resistance Standards . . . . 73

(8.) Comparisons at the National Physical Laboratory . . . . 75

(9.) Hysteresis in Manganin Coils . 77

C. Current.

* (10.) The Electrodynamometer . 81

(11.) Duplex Scale Reading . 82

(12.) The Bifilar Suspension . 83

(13.) The Mean Radius of the Large Coils . 83

(14.) Distance between the Mean Planes of the Large Coils . . . 84

(15.) Area of Windings of the Small Coils . 84

(16.) Ratio of the Currents in the Coils . 85

(17.) The Electromotive Force of the Clark Cell . 86

* Now Professor of Physics at the Royal College of Science, South Kensington, London, S. W.

(313.) 11.8.02

56

PROFESSOR HUGH L. CALLENDAR ON

Part III. — Electrical Thermometry.

Page

(18.) The Compensated Resistance Box . . . 87

(19.) Heating of the Thermometers by the Measuring Current . 92

(20.) Ice-point Apparatus . 94

(21.) Insulation of Thermometers . 95

(22.) Differential Measurements . 96

(23.) Reduction of Results to the Hydrogen Scale . 97

Part IV. — Calorimetry.

(24.) Temperature Regulation . 102

(25.) Preliminary Experiments on the Specific Heat of Mercury . 104

(26.) Method of Determining the True Mean Temperature of Outflow . . 105

(27.) Design of the Water Calorimeter . 107

(28.) Improvements in the Design of the Calorimeter . . . 108

(29.) Effect of Variation of Viscosity . 109

(30.) Radial Distribution of Temperature in the Fine Flow-Tube . 110

(31.) Electrical Method of Measuring the Thermal Conductivity of a Liquid . 112

(32.) Superheating of the Central Conductor . 114

(33.) Methods of Eliminating Stream-Line Motion . 116

(34.) Correction for Variation of Temperature-Gradient in the Flow-Tube . 121

(35.) Application to the Mercury Experiments . . 123

(36.) Correction of Results with the Water Calorimeter . ... 125

(37.) Variation of Gradient-Correction with Temperature . 128

Part V. — Discussion of Results.

(38.) Meaning of the Term “ Specific Heat.” . 130

(39.) Choice of a Standard Temperature for the Thermal Unit . 131

(40.) Choice of a Standard Scale of Temperature . 133

(41.) The Work of Regnault . 134

(42.) The Work of Rowland . 136

(43.) The Method of Mixture. Ludin . 137

(44.) The Work of Miculescu. . 138

(45.) The Work of Reynolds and Moorby . 139

(46.) Empirical Formulae . . 141

(47.) Theoretical Discussion of the Variation of the Specific Heat . 144

List of Illustrations.

Fig. 1. Diagram of Thomson-Varley Slide-Box . 63

„ 2. Compensated Resistance Box . 90

,, 3. Ice-Point Apparatus . 94

,, 4. Hermetically-sealed Thermometers . 95

„ 5. Heater, Circulator, and Regulator . 102

„ 6. Diagram of Mercury Calorimeter . 104

CONTINUOUS ELECTRIC CALORIMETRY.

57

Part I. — Introduction.

(1.) General Account of the Origin ancl Progress of the Investigation.

The method of Continuous Electrical Calorimetry, described in the following
paper, was originally devised as part of a Fellowship Dissertation on applications of
the platinum thermometer, at Trinity College, Cambridge, in the year 1886, but, on.
account of unforeseen difficulties, the experiments did not at that time get beyond
the preliminary stage. In the first rough apparatus, a steady flow of water, passing
through a tube about 30 centims. long and 3 millims. in diameter, was heated by an
electric current in a fine spiral of platinum wire of about 5 ohms resistance, nearly
fitting the tube. The steady difference of temperature between the inflow and the
outflow was measured by a pair of delicate mercury thermometers, which it was of
course intended to replace in the final apparatus by a differential pair of platinum
thermometers. The electrical energy supplied was measured by the potentiometer
method in terms of a set of 5 Clark cells and a large German-silver resistance
of 5 ohms in series with the platinum spiral. The potentiometer was specially made
for the work, and consisted of a metre slide-wire, and ten resistances, each equal to
the slide-wire, for extending the scale so as to secure sufficient accuracy of reading.
This potentiometer was still in existence at the Cavendish Laboratory in 1893.
The set of 5 Clark cells were tested by Glazebrook and Skinner (‘ Phil.
Trans.,’ A, 1892), and were still in good condition at a later date. The external heat-
loss in these experiments was found to be much larger than had been anticipated,
and so variable that the results were of little or no value. In order to remedy this
defect, I designed the vacuum-jacket, which was suggested by some experiments of
Sir William Crookes (£ Eoy. Soc. Proc.,’ vol. 31, 1881, p. 239), which appeared to
indicate that the rate of cooling of a mercury thermometer in a very good vacuum
was ten to twenty times less than in air. I therefore regarded the vacuum-jacket as
a most essential part of the experiment, and expected a great improvement to result
from its use. Unfortunately I failed to make the jacket for want of sufficient skill
in glass-work, and abandoned the experiment for the time, until my appointment as
Professor of Physics at* McGill College, Montreal, gave me greater facilities for
carrying out the work. Eventually it proved that the effect of the vacuum-jacket in
diminishing the external loss of heat was not nearly so great as I had been led to
imagine, but it possessed several advantages as a heat insulator over such materials
as cotton wool or flannel. The thermal capacity of a vacuum being negligible, the
time required for attaining a steady state was much shortened. Moreover there was
no risk of error from damp, which is the worst drawback of ordinary lagging.

I had not originally intended to employ the electrical method for determining the
variation of the specific heat of water, but only for comparing the electrical and
thermal units at ordinary temperatures. In the meantime the work of Griffiths,

VOL. CXCIX. - A.

I

58

PBOFESSOK HUGH L. CALLENDAIt OX

with which I was intimately acquainted, had shown that the electrical units were
probably in error, and appeared to indicate a smaller rate of variation of the specific
heat than that given by Rowland. In reconsidering the problem, in 1893, I
therefore determined to attempt the absolute measurement of the ohm and the
Clark cell, in addition to the variation of the specific heat of water over as wide
a range as possible. The method of steady -flow calorimetry appeared to be
particularly adapted to the latter object, as it afforded much greater facility than
that of Griffiths or Rowland in varying the conditions of experiment over a wide
range. For the absolute measurement of the ohm, I immediately obtained estimates
for a Lorenz apparatus of Professor Y. Jones’ pattern, which was eventually ordered
in October, 1894, and is briefly described in Section 6 of this paper. For the
absolute measurement of the Clark cell in terms of the ohm, after spending some
time in designing various forms of electrodynamometer, I decided to employ the
British Association pattern, with certain modifications, which are explained below,
Sections 10 to 16. At the same time I commenced a series of investigations
into the defects of the form of Clark cell described in the Board of Trade
Memorandum, in which 1 was assisted by Mr. H. T. Barnes. This work included
an accurate determination of the variation of the E.M.F. with temperature and
with strength of solution, in addition to measurements of the solubility of zinc
sulphate and of the density of its solutions. It extended further than I had at first
anticipated, and was not completed till the summer of 1896. The results were
published in the ‘Proceedings of the Royal Society,’ vol. 62, pp. 117-152.

In the meantime I had been engaged, during the winter of 1895 and the summer
of 1896, in testing various methods of temperature regulation, and in studying the
theory of the flow of water in fine tubes under the conditions presented by the
proposed method of calorimetry. This was a most important part of the work,
as the determination of the variation of the specific heat over a large range of
temperature exacted great accuracy of regulation, and close attention to details of
design. The method of regulation and circulation finally adopted may appear very
simple and obvious, but it was not reached wfithout considerable expenditure of time
and thought. The experiments on the flow of water heated by an electric current
(Section 33) threw some light on the causes of failure of the rough preliminary
experiments, and supplied the data necessary for the design of the glass-work of the
calorimeter and vacuum-jacket, which was ordered of Messrs. Muller, in Bonn, early
in October, 1896.

At this stage of the investigation, finding that I should not have sufficient leisure
during the work of the session to carry out the research single-handed, as I had at
first intended, I secured the assistance of Mr. Barnes, who had already proved his
ability in the making and testing of Clark cells. Our first experiments were made
on mercury, which, being itself a conductor of electricity, presented fewer difficulties
than water, The wafer apparatus was fitted up and tested shortly before the

CONTINUOUS ELECTRIC CALORIMETRY.

59

meeting of the British Association, in 1897, but it was at that time incomplete in
certain important details, and only three sets of observations, at 5°, 25°, and 45°,
were obtained. At the commencement of the next session I secured the services of
Mr. Stovel, the most promising of the electrical students of the previous session, to
assist Mr. Barnes in setting up the apparatus and taking the observations. I spent
a good deal of my leisure at this time in the adaptation of the method to the
determination of the specific heat of steam, but continued to give the closest personal
supervision to the work on the specific heat of water, and made several tests of the
apparatus in the vacations when I had more leisure. A great part of the work
during this session consisted in perfecting the mechanical details of the apparatus,
which is always a most important and laborious process in an investigation of this
character. The last work in which I personally assisted before leaving Montreal
was the drawing and annealing of the platinum- silver wire for the Mica Current-
Standards referred to in Section 7. By this time the fundamental portions of
the apparatus had been practically perfected, but the observations, though very
numerous, did not extend beyond the range 0° to 55°, and they had for the most
part been taken for the purpose of testing improvements which from time to time
were introduced, and could not be regarded as parts of a regular series.

When I left Montreal about the end of May, 1898, it was arranged that Mr. Barnes
should continue the experiments throughout the summer, and should follow me to
England with the apparatus as soon as I could make preparations for carrying on the
work in my newT laboratory. Unfortunately this plan proved to be impracticable,
which caused some delay in the work, as I was unable to render him any material
assistance by correspondence at such a distance, owing to the impossibility of
detecting sources of error in any particular case without seeing the apparatus or the
observations. But by the end of the McGill College session in April 1899, he had
succeeded so well in overcoming his difficulties, and the work appeared to be
progressing so favourably, that it seemed inadvisable to disturb the apparatus. I
therefore reluctantly consented to abandon any further share in the observations.
It had originally been intended that 1 should write the paper describing the theory
and results of the investigation ; but as, in the end, Mr. Barnes was solely
responsible for the final series of observations, it seemed more appropriate that he
should write the account of that part of the work.

The primary object of my own contribution is to supplement his account of the
final observations by a general discussion of the theory of the experiment, and a
description of the difficulties encountered in the earlier stages. He was unable to
speak with authority on these points, as a good deal of this work was done before he
joined the investigation, and I had not thought it necessary to give him a detailed
account of it, since it was originally intended that we should finish the work
together. A similar partition of authorship has already been sanctioned in a similar
case in the work of Beynolds and Moorby, and possesses undoubted advantages in

60

PROFESSOR HUGH L. CALLENDAR OX

presenting the results from two distinct and independent points of view. It was the
more necessary in the present instance owing to the comparative independence of
our several shares in the work, and to the impossibility of satisfactory collaboration
at such a distance. I had hoped at one time that it might be possible by some
rearrangement of the matter to weld the separately written portions into a
continuous whole, but as the part written by Dr. Barnes had already been accepted
by the Royal Society, and the Abstract had been already published, it appeared
desirable that it should be printed without alteration as nearly as possible as it was
received, subject only to a rearrangement of the Tables of Results, and the addition
of one or two samples of the original observations.

The delay in publication has been partly due to the necessity of this rearrange¬
ment, and partly to the difficulty of obtaining satisfactory determinations of the
resistance of the manganin standard ohm, on which the absolute values of the results
depended. I have taken advantage of this delay to verify the calculations as far as
possible, and to subject the whole work to as complete and careful a revision as the
time at my disposal would allow. The final results do not materially differ from
those previously published in the ‘Report of the British Association, Dover,’ 1899,
and in the ‘Physical Review.’ There was, therefore, no need for haste so far as the
numerical results of the work were concerned, but it was important in an investiga¬
tion of this character that all the details of the apparatus, and the theoretical and
practical difficulties of the work should be adequately explained and illustrated.

[Added March 1 1th, 1903. — Frequent references are made in the following pages
to the paper by Dr. Barnes, infra, pp. 149-263, describing his experimental results.
These references are generally indicated by the name (Barnes) in brackets, with
the addition of the page, table, or section referred to.

It is hardly necessary that I should say anything here in praise of the con¬
scientious accuracy with which Dr. Barnes has carried out his share of the work.
In re-arranging the tabular summary of observations (Barnes, Table XVIII., p. 243), I
have endeavoured to indicate clearly the order of accuracy attained, and it must be
evident to anyone who studies the paper, that it would be difficult to make any
improvement in this respect.]

Part II. Electrical Measurements. — (A.) Potential.

(2.) Adrantajes of the Potentiometer Method .

The simplest method of observing the electrical energy expended in the calori¬
meter would be to measure the current C, and to assume the value of the resistance
It to be that corresponding to the observed mean temperature of the calorimeter,
fhe watts expended woidd then be given by the formula, W = C~R. This method was
adopted by the majority of the earlier experimentalists. An equally simple method,

CONTINUOUS ELECTRIC CALORIMETRY.

61

but more seldom practised, would be to observe the difference of potential E on the
conductor, assuming the resistance as before. The advantage of assuming the
resistance is that one reading only is required, but, as Rowland pointed out, the
temperature of the resistance, when the heating current is passing through it, must
be considerably higher than that of the calorimeter. This would introduce a serious
error, unless it were possible to use a wire of some material like manganin, in which
the variation of resistance with temperature could be neglected.

Griffiths (‘ Phil. Trans.,' A, 1893) adopted the method of balancing the potential
difference on the conductor against a number of Clark cells in series, and deduced
the expenditure of energy in watts from the formula W = E~/R, by assuming the
value of the resistance. He tried manganin to avoid the error of super-heating, but
found that it was not sufficiently constant. In the end he found himself compelled
to use platinum for the conductor, but avoided the error due to super-heating by
measuring the actual excess-temperature of the wire as nearly as possible under the
conditions of the experiment.

Schuster (‘ Phil. Trans.,’ A, 1895) adopted the same method of balancing the P.D.
on the conductor, but did not assume the value of the resistance. Instead of this,
he measured the time integral of the current with a silver voltameter. This is a
theoretically perfect and most appropriate method of procedure, but it introduces an
additional measurement, and limits the accuracy to that attainable with the silver
voltameter.

For our method of experiment there were several objections to the use of the
silver voltameter, which put it practically out of the question. As is well known,
when the current is first turned through the voltameter, the resistance changes
considerably for some time. This makes it difficult to keep the P.D. on the
conductor accurately balanced against the Clark cells unless the whole resistance
in circuit is large. A. change of this kind in the current at the moment of starting
the experiment would be a fatal defect in the steady-flow method of calorimetry,
as it would disturb all the temperature conditions, which must be perfectly steady
and constant before observations are commenced. Moreover, it happened to be
most convenient for our purpose to employ currents from 5 to 10 amperes, which
would require very large voltameters, and could not be continuously regulated
without constructing special rheostats. In any case regulation by hand would
involve some discontinuity in the heat-flow, which it was desirable to avoid. We
found it best not to make any attempt to control the current artificially, but to
employ very large and constant storage cells, and to compensate the slow rate of
running down of the current by the running down of the head of water, so
that the temperature-difference might remain practically constant throughout the
experiment.

Besides the above special objections to the use of the silver voltameter, there are
the general objections : (1) that the voltameter method gives only the time-integral

62

PROFESSOR HUGH L. CALLENDAR ON

of the current, and does not permit the course of variation of the current to be
accurately followed throughout an experiment ; (2) that with Clark or cadmium cells
of a suitable pattern it is possible to attain an order of accuracy in the relative values
of the readings about ten times as good as that attainable with a silver voltameter.
It was most important for our purpose to obtain accurate relative values, and what¬
ever might be the doubt as to the absolute values of the E.M.F. of the cells, there
could be none as to their constancy, which was easily tested over considerable periods
of time.

It therefore appeared most satisfactory to measure the current by passing it
through a suitable resistance, and observing the P.D. on the terminals with a
potentiometer in the usual manner. The introduction of the potentiometer may
appear at first sight to be an additional complication and source of error ; but it
really made the observations much simpler, and I satisfied myself by careful tests of
the instrument that an accuracy of 1 in 100,000 was readily attainable so far as the
potentiometer readings were concerned. Besides, it was unnecessary, with the
potentiometer, to keep the P.D. on the conductor balanced against an integral
number of cells, and it was, therefore, possible to adjust the electric current to give
the same rise of temperature with different values of the flow of liquid. This most
essential adjustment could not be so conveniently or quickly effected by varying the
How of liquid as by regulating the electric current with a low-resistance rheostat. It
also proved in practice to be much more convenient to take all the electrical readings
on a single instrument, instead of having the silver voltameter as well as the
potential balance to attend to.

It will be seen that our method is independent of any assumption with regard to
the electrochemical equivalent of silver, although the contrary is apparently assumed
in discussing our result both by Ames (‘Report to the Paris Congress of 1900 on the
Mechanical Equivalent of Heat’), and by Griffiths (‘Thermal Measurement of
Energy,’ p. 93). The method of measurement is ultimately equivalent to that of
Griffiths, as it makes the result depend on the International Ohm, and the Clark
cell. The measurement of the current by observing the P. D. on a known resistance,
when combined with the observation of the P.D. on the heating conductor itself, is in
effect equivalent to the measurement of the resistance of the heating conductor
under the actual conditions of the experiment, in the most direct manner possible.
The watts expended are derived from the formula E2/R, so that an error in the
absolute value assumed for the standard cell is twice as important as an error in the
value of the ohm.

If x0 is the balance-reading of the potentiometer when the standard cells are
connected, and e the E.M.F. of the standard cells, and if x , x" are the readings
corresponding to the P.D. on the heating conductor and the standard resistance S
respectively, the expression for the heat-supply in watts is evidently

EC = c*x’ x"/x* S.

CONTINUOUS ELECTRIC CALORIMETRY.

63

The accuracy of the reading xQ for the cells, which enters by its square, is twice as
important as that of x' or x", but it was also more easy to obtain with certainty,
since the cells were kept at a constant temperature, and the reading xQ seldom
changed by more than 1 in 50,000 in the course of an experiment. It will be seen
that, for the determination of the variation of the specific heat, the most important
point in the electrical measurements is the question of the accuracy of calibration of
the potentiometer, which is described in the following sections. The absolute values
of the units are less important, but I have added a brief account of experiments on
the absolute value of the Clark cells, and of the tests of the standard resistances, as
they possess an interest of their own, even apart from the question of the absolute
value of the “mechanical equivalent.”

(3.) Description of the Potentiometer.

The form of potentiometer selected as being most convenient for the purpose was
the well-known Thomson- Varley Slide-Box, which is described and figured in many
electrical works (e.g., Munro and Jamieson’s “ Pocket-Book,” p. 150). The annexed

figure shows the arrangement of the connections, and will be useful for reference in
explaining the details of the calibration.

The main dial ABCD contains 101 coils, each of 1000 ohms resistance, connected
in series, the ends of the series being connected to the terminals AB. The ends of
each coil are connected to platinized studs, which are indicated by the black dots in
the diagram. A pair of revolving contact springs, fixed to an ebonite handle, travel
round the dial. These springs are severally connected to the terminals C and D, and
the distance between them is adjusted so that they bridge over two of the 1000-ohm
coils of the main dial.

The second, or “Vernier” dial, CfD'G, consists of a series of 100 coils of 20 ohms
each, the ends of which are connected to C' and D', and are thus, by way of C and D
and the double revolving contact of the main dial, in parallel with two of the
1000-ohm coils of the main dial. Since two coils of the main dial are always shunted
in this manner by the vernier dial, the effective resistance between C and D is
reduced to 1000 ohms, and the whole resistance of the potentiometer between the

64

PROFESSOR HUGH L. CALLENDAR ON

terminals A and B, to 100,000 ohms. The ends of each of the 20-ohm coils of the
vernier dial are connected to platinized studs arranged in a circle, which make
contact one at a time with a single revolving contact spring, connected to the
galvanometer terminal G. This arrangement of main and vernier dials permits the
sub-division of each hundredth part of the whole resistance into one hundred parts,
so that the reading of the two dials gives the P.D. to be measured directly to one
part in ten thousand of the P. D. on the terminals AB.

The advantages of this form of potentiometer, in addition to its high resistance,
were (l) the great facility and rapidity of reading and manipulation, and (2) the
symmetry of construction, which permitted a very high order of accuracy of calibration
to be attained, and greatly facilitated the application of corrections, as compared
with the usual type of instrument in which a bridge-wire is employed for the finer
sub-divisions.

In the use of this instrument in our experiments the P.D. to be measured seldom
exceeded 4 volts. The terminals AB were permanently connected to three Leclanche
cells, which gave a very steady current through so high a resistance. The reading of
the two Clark cells employed as a standard varied by a few parts in 10,000 only from
week to week, and generally remained constant to 1 in 100,000 for the short interval
of 1 5 minutes corresponding to any single experiment.

The galvanometer employed with this potentiometer had a resistance of
110,000 ohms. The astaticism of the needles was adjusted as carefully as possible, so
that the effect of disturbance of earth-currents due to the electric railway might be
negligible. The suspended system was fitted with a very perfect mirror and a
damper to make it practically dead-beat. The sensitiveness was adjusted by control
magnets to give a deflection of approximately 10 scale-divisions for one division of
the vernier dial (1 in 10,000). The perfection and steadiness of the image was such
as to permit reading to a small fraction of a scale-division. The first four figures of
the reading were given by the setting of the dial contacts. It wras easy to estimate
the fifth figure at any moment by inspection of the galvanometer deflection. The
temperature conditions wTere generally so steady in the course of an experiment, and
the diminution of the electric current and the water-flow so gradual and regular, that
it was possible, as a rule, to predict the reading of the P.D., either on the standard
resistance or on the heating conductor, to 1 in 100,000 for at least five minutes
ahead.

As there were no observational difficulties to contend with in the electrical
readings, the relative order of accuracy of the results would be limited only by the
constancy of the Clark cells and the current standards, and by the order of accuracy
attainable in the calibration of the potentiometer and in the permanence of the
relative values of the coils. The coils of the main dial, which were the most
important, were all precisely similar, wound with the same wire and carefully
protected from sudden or unequal changes of temperature. The ratio of the

CONTINUOUS ELECTRIC CALORIMETRY.

65

resistances of the two halves of the dial was frequently checked with consistent
results, as a precaution to give warning of any accidental flaw. This precaution was
by no means superfluous, for on one occasion in November, 1897, a fault, amounting
very nearly to complete rupture of the wire, was discovered by Mr. King in this
manner. It was however easily located and rectified without altering the relative
values of the coils.

(4.) Method of Testing.

The method of testing the ratio of the two halves of the slide-box was as follows : —
The slide-box was connected by the terminals AB in parallel with a 100,000-ohm box
consisting of ten coils of 10,000 ohms each. The battery was connected as usual at
A and B, one terminal of the galvanometer to G, and the other to the middle of
the 100,000-ohm box. The slide-box contacts were set at 50,000 ohms. To take
one particular experiment as an example, the deflection of the galvanometer observed
on reversing the battery was 77 scale-divisions. When the contact was set at
50,010 ohms, the deflection was 215 scale-divisions in the same direction, showing a
sensitiveness of 138 scale-divisions on reversal for a change of 1 in 5000 in the
reading. The contact was then set back to 50,000, and the two halves of the box
were interchanged with respect to the rest of the circuit by interchanging the
connections at A and B. The deflection observed was increased from 77 to 181
scale-divisions in the same direction as before. The effect of interchanging the two
halves is the same as if the slider were shifted through a resistance equal to
their difference. Hence the difference of the two halves is to 10 ohms as 104 is to
138. The first half of the box is evidently the smaller, as the effect of interchanging
is the same as that of increasing the reading. The correction to be applied to the
reading, to reduce to mean ohms of the box, is half the difference of the two halves,
and is negative, since the first half is the smaller. We have, therefore,

Correction at reading 50,000 ohms = — 10 X 104/2 X 138 = — 3-8 ohms.

The galvanometer deflections in each case were observed several times and the mean
taken. The details were also varied by using different resistances for the ratio arms
in the comparison and different galvanometers. Observations were taken by different
observers at various temperatures on several occasions, at intervals during five years.
The greatest divergence of the results from the mean value is less than 1 part in
100,000 (’4 ohm in 50,000 ohms), which is strong evidence that the relative values
of the corrections at any part of the box could be relied on at any time to a similar
order of accuracy.

The following is a summary of the results of all the tests of which full details
have been preserved, but several other tests were made from time to time as a
precaution : —

VOL. cxcix. — A.

K

66

PROFESSOR HUGH L. CALLENDAR ON

Table I. — Verification of Correction at Middle Point of Slide-Box.

Date.

Observers.

Correction
(ohms in 50,000).

February, 1894 . . . .

C ALLEND AK.

-3-8

December 20th, 1894 . . .

Callendar.

-3-96

January 29th, 1895 . . .

King.

-3-77

November 24th, 1896.

Thomson and Stovel.

-4-2

February 2nd, 1897 .

Blair and Macdonald.

-3-72

March 4th, 1897 ....

Pitcher and Edwards.

-3-50

April 22nd, 1898 ....

Stovel.

-3-43

January 27th, 1899 . . .

Barnes.

-3-95

Some of the above observations were taken by fourth-year students in the course
of their work, but in the majority of cases I personally verified the readings and
results at the time of entry.

( 5 . ) Me th ocl of Calibration.

In the calibration of the slide-box, the point of most importance was to determine
the correction for each reading of the main dial, i.e., at 100 equidistant points of the
whole range. The vernier dial was so small in comparison that the errors of its
individual coils were negligible in their efiect on the whole reading, although it was
necessary at each point to take account of the difference of resistance of the whole
vernier dial and the pair of coils shunted by it in any position of the slider.

After several trials of various methods extending over nearly a month, I came to
the conclusion that the most convenient and accurate method of performing the
calibration was to determine the relative values of the coils of the main dial in pairs
by comparison with the 2000 ohms of the vernier dial. The flexible copper cable
connecting the terminals D and D' was disconnected, and the terminals were
connected to a galvanometer and to a pair of exactly similar resistances of 2000 ohms
each forming the ratio arms P and Q of a Wheatstone bridge, the other two arms of
which were the vernier dial S and any pair of consecutive coils R„ and R„ + 1 of the
main dial. A battery of two storage cells, selected for constancy, was connected to
the point between the ratio arms and to the terminals CC'. The deflection d„ of the
galvanometer corresponding to any setting of the slider was proportional to the
difference of the sum of the corresponding pair of coils R„ and R„ + 1 of the main dial
from a unit SP/Q, which was approximately 2000 ohms, and remained constant
throughout the comparisons. The . value of this deflection was reduced to ohms by
observing the change of galvanometer deflection s produced by a change of 1 ohm in
one of the arms. This observation was repeated at intervals during the calibration.

CONTINUOUS ELECTRIC CALORIMETRY. b 7

The advantage of this particular arrangement was partly that of expedition and
convenience, partly that of avoiding systematic errors due to changes of condition or
temperature while the calibration was proceeding. The construction of the vernier
dial, 100 coils of 20 ohms each, made it a good standard of comparison, as there was
no risk of appreciable heating from the current employed, although it was necessarily
kept on for more than an hour. Moreover, as it was constructed of similar wire and
enclosed in a similar box to the main dial, it was probable that any change of the
surrounding conditions of temperature would affect the two similarly. The heating
effect of the current on P and Q would be sufficiently eliminated by their similarity
of construction.

Readings taken in this manner, with the slider set in each position of the main
dial, gave 100 equations of the following form : —

R« + R«+i = SP/Q + dn/s . (1).

To determine the correction at each point of the main dial, and the relative values
of the 101 resistances and the vernier dial, it was also necessary to determine the
ratio of any two of the coils to each other, and the ratio of the two together to the
vernier dial. This was effected by the method of interchanging, as already described
for determining the ratio of the two halves of the slide-box.

The ratio of coils Rt and R., to the vernier dial S was found to be

(R2 + R2)/S = P000039.

The ratio of coils R: and R2 to each other was found to be

Pvj/Ro = P000400.

In the latter case the galvanometer contact was made by means of a copper wire
to the stud between 1 and 2, the glass cover being removed for the purpose of
this test.

The observation of the values of the deflections d for the 100 equations of the
type (1), was repeated on two sejjarate occasions. On the first occasion the
110,000-ohm galvanometer was employed, but it was found that when the galvano¬
meter was adjusted to a suitable degree of sensitiveness for the experiment, its time
period was too slow, and its zero not sufficiently constant to give the best results.
It took upwards of an hour to obtain the first fifty observations. This series was
not therefore continued throughout the box, but the observations were reduced to
mean ohms of the box by reference to the value of the correction at the middle
point of the box obtained from a separate observation. On the second occasion the
110,000-ohm galvanometer was replaced by one of 2000 ohms resistance, which
was better suited for this particular exjDeriment, though not so well adapted for
observations in which the whole box was employed. The sensitiveness of this

k 2

68

PROFESSOR HUGH L. CALLENDAR ON

galvanometer was adjusted to give a deflection of 167 scale-divisions on reversal for
a change of 1 ohm in 2000 with a time period of 5 seconds, and remained constant
to less than one scale-division throughout the test of the whole box, which occupied
only an hour and a hall.

The observations and results of the two calibrations for the first half of the slide-
box are compared in the following table. The first column contains the reading of
the slider on the main dial. The second column the observed deflection of the
galvanometer cl in equation (l) reduced to ohms by dividing by s. Since s = 167 in
the second series, one unit in the second decimal place of d/s corresponds to nearly
2 scale- divisions deflection observed. The observations were taken to half a scale-
division, but owing to slight variations of sensitiveness and zero it was not considered
worth while to work the values of d/s beyond the nearest hundredth of an ohm.
The next column gives the error c?R in ohms of each separate resistance of the main
dial in terms of the mean of the whole, deduced from equations (1) by the aid
of (2) and (3). The fourth column gives the correction dn in ohms to the reading at
each point. This correction is equal to the sum of the errors of all the coils up
to the point considered, subject only to a small correction, called the “ vernier-
correction.” to allow for the fact that the next two coils are shunted by the vernier

J

dial. The value of the vernier-correction is given by the following expression —

Vernier-correction to reading n = — n (‘38 — 3cZR;i+1 — 3c/R„+;,)/400.

This correction is often negligible when n is small, but sometimes reaches '5 or '6
of an ohm near the higher readings. The next three columns in the table give the
corresponding values of the same quantities d/s. eTR, and dn, deduced from the
readings taken during the second calibration. Comparing the two sets it will be
observed that the discrepancy very rarely exceeds half an ohm, which is only one
part in 200,000 of the whole resistance.

CONTINUOUS ELECTRIC CALORIMETRY.

69

Table II. — Calibration of 100,000-ohm Slide-Box. Corrections in ohms.

Reading,
of main
dial n.

Observations, Series I.
November 24, 1894, at 15°-5 C.

Observations, Series II.
December 20, 1894, at 20°’2 C.

Difference
of Series

I-II.

d/s.

dR.

Correction.

d/s.

dli.

Correction.

0

+ 1

04

0

+ 1

26

0

0

1

+ 0

52

+ 0

43

+ 0

43

+ 0

55

+ 0

•45

4-0-45

-0

02

2

+ 0

26

+ 0

05

+ 0

45

+ 0

20

4-0

•05

4-0-49

-0

04

3

+ 0

64

-0

10

+ 0

36

+ 0

77

-0

•26

4-0-24

4-0

12

4

+ 0

92

-0

34

+ 0

03

+ 1

11

-0

•30

-0-07

4-0

10

5

+ 0

40

+ 0

39

+ 0

40

+ 0

54

+ 0

•31

+ 0-24

+ 0

16

6

+ 1

00

-0

06

+ 0

36

+ 1

21

4-0

•04

4-0-30

+ 0

06

7

+ 1

42

-0

13

+ 0

26

+ 1

53

-0

•25

+ 0-07

+ 0

19

8

+ 0

68

+ 0

54

+ 0

76

+ 0

74

+ 0

71

4-0-74

4-0

02

9

+ 0

40

+ 0

29

+ 1

03

+ 0

51

4-0

06

+ 0-78

+ 0

25

10

+ 0

52

-0

20

+ 0

83

+ 0

62

-0

08

4-0-71

+ 0

12

11

+ 0

50

+ 0

01

+ 0

84

+ 0

58

-0

17

+ 0-53

4-0

31

12

+ 1

56

-0

08

+ 0

86

+ 1

70

4-0

03

+ 0-66

+ 0

20

13

+ 1

50

-0

01

+ 0

85

+ 1

72

-0

21

4-0-46

4-0

39

14

+ 0

46

+ 0

98

+ 1

72

+ 0

51

+ 1

16

+ 1-50

4-0

22

15

+ 0

44

-0

07

+ 1

65

+ 0

52

-0

19

4-1-30

4-0

35

16

+ 0

26

-0

06

+ 1

57

4-0

37

-0

06

+ 1-22

4-0

33

17

+ 0

32

-0

09

+ 1

48

+ 0

40

-0

18

4-1-04

4-0

44

18

+ 0

22

-0

24

+ 1

22

+ 0

31

-0

21

4-0-82

4-0

40

19

+ 0

04

-0

03

+ 1

16

+ 0

15

-0

15

4-0-64

4-0

52

20

+ 0

62

-0

34

+ 0

91

+ 0

84

-0

30

+ 0-44

4-0

47

21

+ 0

90

-0

21

+ 0

74

+ 1

11

-0

30

4-0-18

4-0

56

22

+ 0

68

+ 0

24

+ 0

94

4- 0

94

4-0

39

+ 0-54

4-0

40

23

+ 0

32

+ 0

07

+ 0

95

+ 0

52

-0

04

+ 0-43

+ 0

52

24

+ 0

14

+ 0

02

+ 0

94

4-0

26

4-0

22

+ 0-60

4-0

34

25

+ 0

16

-0

29

+ 0

65

+ 0

30

-0

46

+ 0-14

4r0

56

26

+ 0

26

-0

16

+ 0

50

+ 0

45

-0

04

+ 0-13

4-0

37

27

+ 0

30

-0

27

+ 0

24

4-0

58

-0

42

-0-27

4-0

51

28

-1

04

-0

06

-0

11

-0

84

4-0

12

-0-45

4-0

34

29

-0

96

-0

23

-0

34

-0

74

-0

29

-0-73

4-0

39

30

+ 0

42

- 1

40

- 1

44

4-0

71

- 1

31

-1-72

+ 0

28

31

+ 0

30

-0

15

- 1

62

+ 0

52

-0

19

-1-96

4-0

34

32

+ 0

28

-0

02

-1

65

4-0

35

4-0

14

- 1-86

4-0

21

33

+ 0

16

-0

27

- 1

95

4-0

29

-0

38

-2-26

+ 0

31

34

+ 0

46

-0

04 •

- 1

92

+ 0

67

-0

03

-2-20

+ 0

28

35

+ 0

36

-0

39

-2

34

4-0

62

-0

43

-2-64

+ 0

30

36

+ 0

20

+ 0

26

- 2

12

4-0

41

+ 0

35

-2-35

+ 0

23

37

+ 0

12

-0

49

- 2

63

4-0

32

-0

49

-2-87

4-0

24

38

+ 0

06

+ 0

10

-2

56

4-0

27

4-0

14

-2-75

+ 0

19

39

+ 0

80

-0

57

-2

92

+ 0

99

-0

58

-3-12

+ 0

20

40

+ 0

92

+ 0

04

- 2

84

+ 1

07

4-0

09

-3-00

+ 0

16

41

+ 0

82

+ 0

17

- 2

70

4-0

98

4-0

14

-2-89

+ 0

19

42

+ 1

20

+ 0

16

— 2

42

4-1

38

+ 0

18

-2-58

+ 0

16

43

+ 0

54

+ 0

07

_ 2

56

4-0

69

4-0-

05

-2-75

+ 0

19

44

+ 0

08

+ 0

54

_ 2

17

4-0

24

+ o-

57

-2-33

+ 0

16

45

+ 0

16

-0

59

_ 2

74

4-0

29

-o-

64

-2-96

+ 0

22

46

+ 0

22

+ 0

08

-2

64

+ 0

41

4-0-

12

-2-80

+ 0

16

47

+ 0

64

-0

51

-3

01

4-0

84

-o-

59

-3-25

4-0

24

48

+ 0

00

+ 0

14

-3

10

4-0

13

4-0-

24

-3-26

+ 0

16

49

-0

20

-0

09

-3

37

-0

07

-o-

15

-3-49

4-0

12

50

-o-

24

-0

50

-3

79

-0

06

-o-

47

-3-96

+ 0

17

70

PROFESSOR HUGH L. CALLENDAR ON

The following table gives the corrections found for the second half of the sliae-box
on the second occasion, December 20th, 1894.

Table III. — Calibration Corrections of Second half of Slide-Box.

Reading
of main
dial.

Correc¬
tion in
ohms.

Reading
of main
dial.

Correc¬
tion in
ohms.

Reading
of main
dial.

Correc¬
tion in
ohms.

Reading
of main
dial.

Correc¬
tion in
ohms.

Reading
of main
dial.

Correc¬
tion in j
ohms.

50

-3-96

60

-4-58

70

-3-13

80

-1-33

90

-0-92

51

-4-42

61

-4-68

71

-3-32

81

- 1-21

91

-0-96

52

-4-57

62

-4-07

72

-3-19

82

-1-11

92

-0-45

53

-5-09

63

-4*04

73

-3-03

83

- 1-24

93

-0-70

54

-4-88

64

-3-69

74

-3-03

84

-1-38

94

-0-63

55

- 5 • 15

65

-3-60

75

-2-47

85

- 1-70

95

-0-82

56

-5-26

66

-3-62

76

-2-16

86

- 1-83

96

-0-94 |

57

-5-31

67

-3-68

77

-1-92

87

-1-64

97

-0'22

58

-5-19

68

-3-45

78

- 1-61

88

- 1-71

98

-0-23

59

-5-23

69

-3-51

79

-1-64

89

-1-24

99

-0-06

In comparing the differences between the two calibrations given in the last column
of Table II. , it will be noticed that there is a cumulative divergence amounting to
about half an ohm at the middle of the range. This is the kind of error to be
expected in this method of calibration. It might be explained by the considerable
difference in temperature of the box on the two occasions, but it is within the limits
of error of the first series. The galvanometer was not sufficiently steady on that
occasion, and the temperature rose nearly half a degree in the course of the
observations. The observations serve, however, as a satisfactory verification of those
of Series II.

It will he observed that the correction does not amount to so much as 1 part
in 10,000 of the reading at any point of the slide-box, except quite near the
beginning, a part which was never used in accurate comparisons. Also that the
change of the correction in passing from one point to the next, never exceeds
1 part in 100,000 of the reading in the second half of the box, although the
errors of two or three of the individual 1000-olnn coils exceeded 1 ohm. This is
due to the levelling effect of the vernier.

It must be remembered that the corrections were never required beyond the
nearest ohm, so that a difference of less than ‘5 could be neglected.

It might be supposed that greater accuracy of calibration would have been
attained by dividing up the box into subsidiary intervals of 10,000 ohms, and
comparing each of these intervals with an auxiliary resistance, on the analogy of the
method usually employed in the calibration of a mercury thermometer. I did not
find, however, that any advantage was obtained by this procedure, and it is evident,
on reflection, that the two cases are not precisely analogous. The advantage of

CONTINUOUS ELECTRIC CALORIMETRY.

71

employing longer columns in the case of a mercury thermometer is that the errors
of estimation become of relatively less importance. It is practically possible to
measure both the long and short columns to the same fraction of a degree, and an
error of ‘001° in a column of 20° is of much less relative importance than in a
column of 2°. In the case of the resistance box, on the other hand, the relative
accuracy of measurement is undoubtedly greater in the case of the smaller
resistances. A 10,000-ohm coil cannot be measured with the same order of accuracy
as one of 1000 ohms. This is partly due to difficulties of insulation in the winding
of the coils themselves, and partly to the fact that wire finer than 2 or 3 millims.
cannot be drawn and covered satisfactorily. As a consequence, high-resistance
galvanometers are necessarily less efficient than low-resistance instruments of similar
construction. The best high-resistance coils are constructed of a number of lower
resistance coils in series, as in the Thomson- Varley slide-box, which permits a higher
order of insulation than winding in a single coil.

(B.) Resistance.

(6.) 'The Lorenz Apparatus.

Although this apparatus was not actually applied to the direct determination of the
resistances employed in this investigation, owing to delay in delivery, it was originally
ordered with this object, and the preliminary experiments which were made by Professors
Ayrton and Jones in testing the apparatus before it was sent out, are of so great value
as bearing on the absolute value of the ohm that they cannot be passed over without
mention. The null method of Lorenz, in which a resistance is directly determined in
terms of the speed of rotation of a disc spinning in the field of a co-axial coil of known
dimensions, is generally admitted to be the most accurate for the absolute measure¬
ment of resistance. The McGill College apparatus was constructed by Messrs.
Nalder Bros, to my order, under the direct supervision of Professor Yiriamu
Jones, and embodied all the improvements introduced into the method by himself
and by Lord Bayleigh. The most important new feature of the design was the
winding of the coil on a heavy cylinder of marble, instead of metal as employed in
Professor Jones’ original apparatus. This material possesses the advantage of high
insulating properties, great rigidity, and small thermal expansion. The employment
of a marble cylinder made it possible to wind the coil with uncovered wire with the
object of obtaining the most exact measurement of the dimensions, but on account
of some difficulties of insulation, the original winding of bare wire was eventually
replaced by one of silk-covered wire coated with paraffin and shellac varnish.

The results of the tests made by Professors Ayrton and Jones with this coil, at
the laboratory of the Central Institution, during 1896 and 1897, have been published
in the Reports of the British Association for 1897 and 1898, and in the ‘ Electrician.’
They give a value for the Board of Trade standard ohm nearly 3 parts in 10,000

72

PROFESSOR HUGH L. CALLENDAR ON

greater than 10° C.G.S. units. This divergence might be explained as due to
imperfect insulation if the precautions taken had not been so great, but it more
probably represents the order of accuracy at present attainable in the absolute
measurement of resistance by this or any other method. It is hardly possible
that it could be entirely accounted for by errors of measurement of the diameter of
the coil and the disc, although a discrepancy of 1 in 10,000 was actually found
between the calculated and measured diameters of the coil (‘ B.A. Rep.,’ 1897,
p. 217), which points to some uncertainty in this direction.

At the time when I first examined the coil of the Lorenz apparatus, some time
after its arrival in Montreal, while it was being set up, it seemed to me that the wire
had worked a little loose on the marble, owing to some effect of the drying of the
insulating tape and varnish, or to straining of the soft insulated wire due to
contraction on exposure to cold. This would necessarily occur owing to the great
difference in the coefficients of expansion of the wire and the marble, and the very
small limits of elasticity of the soft annealed wire. Professor V. Jones himself,
with whom I discussed the question in September, 1897, shortly after the arrival of
the apparatus in Montreal, considered that the diameter could not be satisfactorily
measured with a silk-covered wire, and strongly recommended the re-winding of the
coil with bare wire. Accordingly, I procured for this purpose a sample of highly
elastic silicium-bronze wire of high conductivity. I satisfied myself that the limits
of elasticity of this wire would be ample, if it were wound on under suitable tension
at a suitable temperature, to keep it perfectly tight on the marble cylinder for any
range of temperature to which it was likely to be exposed. The tightness of the
wire in practice is most important, from the point of view of insulation as well as
from that of accurate measurement of the diameter. If the diameter of the wire is
nearly equal (as it must necessarily he) to the pitch of the screw thread, a very
slight defect in tightness or straightness will produce a short circuit. It is quite a
difficult matter to wind a perfect coil of 200 turns of this size, unless the wire is
highly elastic and quite free from kinks.

Owing to the great importance of securing perfect insulation, I proposed to adopt
the method which I had already put in practice in the case of the electro-
dynamometer, namely, to wind the coil in a double screw thread with two separate
wires, in order to have a check on the perfection of the insulation, which could
be applied at any time after the coil had been wound, or at any moment during
the actual experiments.

In consequence of the delay caused by the failure of insulation of the first coil,
rhe apparatus did not arrive in Montreal until the beginning of September, 1897.
Some time was occupied in the course of the winter in building a suitable pier, and
setting up the apparatus. But when I was about to commence observations, I
received news of my appointment to the Chair of Physics at University College,
London, which made it necessary for me to abandon the work.

CONTINUOUS ELECTRIC CALORIMETRY.

73

(7.) Values of the Resistance Standards.

The currents employed in this investigation were measured on the Thomson- Varley
potentiometer by comparing the difference of potential on the terminals of a specially
constructed resistance, called the Current-Standard, with that of a pair of hermetically
sealed Clark cells. This current-standard consisted of a single 1-ohm coil, or of two
1-ohm coils in parallel, immersed in a well-stirred oil-bath. The resistance coils
employed for this purpose in the earlier experiments were made of thick manganin
wire of the best quality procured from Germany. The diameter of the wire was
1 millim., and the maximum current carried by each single wire was 2 amperes.
The wire was not materially heated above the temperature of the oil, but, as no cooler
was used in the earlier experiments, the temperature of the oil generally rose some
8 or 10° in the course of an hour. This was considered to be of no consequence
with the manganin coils, as they had a temperature coefficient of only -f- '000020 for
ranid changes. On re-testing these coils after some months’ work, it was found that
their resistance had increased by two or three parts in 10,000, and that they
continued to show small variations of this order. It was possible that these changes
might have been due to the solder junctions, which would have explained certain
anomalies observed in the earlier experimental results. It is equally likely, however,
that they were caused by hysteresis in the wire as explained below.

Although the variations of the manganin coils did not exceed a few parts in 10,000,
it was felt that they were quite inadmissible, as the potential readings were taken to
1 in 100,000. For this reason it was decided to make a pair of platinum-silver ohms
wound on mica and annealed at a dull red heat after winding. These will be called
the “ Mica Current- Standards.” They are fully described and illustrated by
Dr. Barnes, p. 173. The method of construction was modelled on that of a platinum
resistance thermometer, and had already been adopted for some years by the
Instrument Company, Cambridge, for the manufacture of standard resistance coils.
Standards constructed on this model possess great constancy, and their temperature
can be determined with accuracy by immersion of the naked wire in an oil-bath. As
compared with manganin coils, they have the disadvantage of a larger temperature
coefficient, but the construction permits the determination of this coefficient with the
greatest certainty, so that it is really no objection to their use for scientific purposes,
for which coustancy is the primary desideratum. The adoption of platinum -silver
necessitated the addition to the oil-bath of a cooling coil of composition tubing, with
a water circulation to keep the temperature steady.

For the determination of the variation of the specific heat of water, the absolute
value of the current-standard was of no moment, but its constancy was of primary
importance. It was also essential to be able to determine the temperature coefficient
accurately, in order to reduce the observations at different temperatures in winter and
summer to the same standard. This was determined under the actual conditions of

VOL. CXCIX. — A.

L

74

PROFESSOR HUGH L. CALLENDAR ON

observation with stirrer and cooler (Barnes, p. 174), by comparison with a
manganin standard ohm, No. 4086, which was kept at a steady temperature. The
value of the temperature coefficient of the mica current-standards is, therefore,
independent of any assumption with regard to the temperature coefficients of the
other standards employed. The low value '000248 of the coefficient found is explained
by the perfect annealing of the wire.

The absolute values of the mica current -standards were referred to the manganin
standard No. 4086. The comparisons of this coil with the mica standards, which
extended over two years, are given in Table IX. (Barnes), and do not show any
relative changes of importance. The observations were taken on twelve different
dates at temperatures varying from 15° to 21° C, the mean temperature of all the
comparisons being 18° '3 6 C.

The temperature coefficient of the manganin standard No. 4086 was assumed to be
•000018 in the reduction of these observations. This value was determined by
observing the change of resistance due to changes of temperature of the oil-bath in
which the coil was immersed, the changes of temperature being effected at the rate of
L0° C. in two or three hours. The results in Table IX. (Barnes) are reduced to a
temperature of 20° C. If, however, we reduce the results to 180,4 C., the mean
temperature of the comparisons, we shall be independent of any assumption with
regard to the temperature coefficients. We thus obtain the values : —

Mica current -standard, No. I. at 1 8°*27 = No. 4086 -f- ’00091 at 18°’4C.

„ II. „ 180,46 = „ „ - -00035 „ „

No. I. and No. II. in parallel at 180,36 C. = Half No. 4086 -p ‘00014 . . (1).

The manganin standard No. 4086 had a Cambridge certificate No. 378 dated 1893,
which stated its value to be ’99978 of a true ohm at 1 5 ’9° C. The preliminary results
for the value of the calorie in terms of the electrical units, communicated to the
British Association in September, 1899, were expressed in terms of this certificated
value as correct. A comparison was subsequently made at McGill College of No. 4086
with a set of 10 platinum-silver standard ohms of the old pattern, the coils of which
were embedded in paraffin wax. These 10 coils all possessed Cambridge certificates,
and their relative values when tested, agreed fairly with those obtained several years
previously, showing no certain indication of change. The mean of the certificates of
these 10 coils gives the value : —

Mean of 10 Pt-Ag standard ohms = ’99962 at 16°’5 C. ... (2).

The details of the comparison of No. 4086 by Dr. Barnes# with 9 of these

* The standard No. 3566 omitted by Barnes happens to be so near the mean of the 10 as to make no
material difference in the result.

CONTINUOUS ELECTRIC CALORIMETRY.

75

standards are given in Table VII. (Barnes). The mean of the results may be stated
as follows :• — -

No. 4086 at 21°"74 C. = Mean of 10 Pt-Ag ohms - -00054 at 21°"59 C. . (3).

The coils were also compared about the same time by Mr. Fraser at a lower
temperature. The value of No. 4086 in terms of the mean deduced from this second
comparison is : — -

No. 4086 at 14c,8 C. = Mean of 10 Pt-Ag ohms -f ’00153 at 1 3°*6 C. . (4).

The details of this comparison are given in Table VIII. (Barnes).

If the mean values of the 10 Pt-Ag ohms at 210,6 and 13°"6 C., respectively, are
deduced from the certificated value of the mean at 1 6°'5 Cl, adopting '000254, the
value taken by Barnes as the temperature coefficient, we obtain the following values
for No. 4086 : —

No. 4086 at 210,74 C. = 1 '00037 ohms (from standards at 21°'6 C.) . (5).

„ „ „ 14°'8C. =1'00041 „ „ „ 13°'6 C.) . (6).

These results are evidently inconsistent with the value + "000018 for the tempera¬
ture coefficient of No. 4086. The individual observations, however, are too consistent
to admit of the supposition of an error of the order of two parts in 10,000 in the
mean of the comparisons. It seems more likely that the value "000254 assumed for
the temperature coefficient of the Pt-Ag standards is too small. In any case, if we
take the mean of the two comparisons, we shall obtain a result which is nearly
independent of the value assumed for the temperature coefficient. We thus
obtain : —

No. 4086 at 18°"4 C. = 1 "00039 ohms (Pt-Ag standards at 17°"6 C.) . (7).

It happens that the temperature 18° "4 C. is precisely that at which the value of
No. 4086 is required for the comparisons with the mica current-standards.

(8.) Comparisons at the National Physical Laboratory.

It appeared practically certain from these comparisons that the value of No. 4086
had increased since the date of the original certificate by nearly "00060 ohm. For
further verification No. 4086 was sent over in a box from Montreal to Cambridge,
and thence to Kew, to be compared with the original standards. Mr. Glazebrook
very kindly undertook the comparison himself, with the following results : —

76

PROFESSOR HUGH L. CALLENDAR ON

Value of No. 4086 from Observations at the National Physical Laboratory.

Date.

Temperature.

Value from 3715.

Temperature.

Value from “Flat.”*

July 20 .

o

25-0

1-00067

0

24-9

1-00067

„ 26 .

24-4

1-00072

24-4

1-00059

,,28 .

21-4

1-00063

21-3

1-00059

,,28 .

21-6

1-00062

—

—

„ 30 .

18-9

1 -00057

—

—

August 1 .

16-0

1-00045

—

—

These observations give a mean value of 1 '00061 ohms at a temperature of
21°‘96 C., and indicate a temperature coefficient of about ‘000027 for changes taking-
place in a few days. This result is inconsistent with the value (5) 1 ‘00037 at
21°‘74 C. found by Barnes, unless we suppose the temperature-coefficient employed
by Barnes for the Pt-Ag coils to be a little too small, or unless No. 4086 had
increased in value on its journey from Canada. Mr. Glaze brook was not perfectly
satisfied with these observations, as the temperature was too high and variable, and
the difference to be measured on the Pt-Ir bridge-wire, as well as the temperature
correction of the standards, was rather large. He therefore repeated the observations
in December, at a lower temperature, under much steadier conditions, with the
following results : —

Value of No. 4086 from National Physical Laboratory Standards.

Date.

Temperature.

Value from 3715.

Temperature.

Value from “ Flat.”

December 13, 1900 .

o

12-1

1-00018

o

11-9

1-00020

,, 18 and 19 .

13-2

1-00016

13-0

1-00018

„ 20 ... .

11-3

1-00016

11-5

1-00015

„ 21 ... .

12-1

1-00017

12-1

1-00015

„ 28 ... .

10-3

1-00010

—

—

The results of these observations give a mean value of 1 '00016 at a temperature of
12°‘0 C. This appeared inconsistent with the July observations. It was also much
lower than the value (6) 1 *0004 L at 14°‘8 found bv Fraser, unless we suppose either
that the coefficient ‘000254 applied by Barnes to the platinum-silver coils was too
small, or that No. 4086 had fallen in value in the interval.

* “ Flat ” is one of the original B.A. unit standard coils belonging to the British Association, the
temperature coefficient of which is taken to be -000277, as determined by Fleming in 1876. 1 ohm is

assumed to be 1 01358 B.A. unit. No. 3715 is a more recent standard ohm.

CONTINUOUS ELECTRIC CALORIMETRY.

77

In order to throw further light on the question, Mr. Glazebrook undertook a
determination of the temperature-coefficient of No. 4086 by heating it up to 20° C.
and allowing it to cool slowly, the case being full of oil. Taking only the morning
observations, which Mr. Glazebrook considers the most reliable, we obtain the
following mean results on the two days : —

Temperature-coefficient of No. 4086 at the National Physical Laboratory.

Date.

Mean temperature.

Mean value from 3715.

January 4, 1901 . .

o

17-2

1-00024

)> 5 ,,

9-4

1-00005

These observations give a mean value of 1 '00015 at 1 3°‘3 C., which agrees very
closely with the December comparisons, but is slightly lower, as though the diminu¬
tion of resistance were still in progress. The value of the temperature-coefficient
deduced is + '000024, which is a very probable value for this kind of manganin for
temperature changes of 10° in one day.

(9.) Hysteresis in Manganin Coils.

From my own experience of the behaviour of manganin, I am inclined to explain
these discrepancies as follows : — The temperature-coefficient of a manganin coil
frequently depends to some extent on the past history and on the rate of heating or
cooling. It may exhibit a kind of lag or hysteresis, the resistance continuing to
increase gradually for some time after a rise of temperature, so that the value of the
coefficient found by short-period observations is often smaller than the value which
applies to changes of long period. Taking the July and December observations at
the National Physical Laboratory as being correct, we find a long-period temperature-
coefficient for No. 4086 of +'000047, which is about twice as large as the short-
period coefficients '000027 and '000024, deduced from the observations from day to
day. It was for this reason that I discontinued the use of manganin for accurate
work in platinum thermometry some years ago, preferring to use platinum-silver coils
annealed at a red heat and compensated for temperature by means of platinum
similarly annealed. The defect is of a kind that would escape notice except in very
accurate work over long periods with coils exposed to considerable changes of
temperature. It also appears to depend on the manner in which the wire has been
annealed.

It is worthy of notice, as a confirmation of this view, that a small effect of this
nature was observed by Marker and Chappuis in their platinum thermometry

78

PROFESSOR HUGH L. CALLENDAR OX

(‘ Phil. Trans./ A, 1900, vol. 194, p. 59), though they do not appear to have appre¬
ciated its full significance. In order to reduce the temperature correction, they
selected a resistance box with manganin coils made by Messrs. Crompton in pre¬
ference to one with platinum-silver coils annealed at a red heat after the pattern
made by the Cambridge Instrument Company. In attempting to determine the
temperature-coefficient of their box by raising the temperature for a short period,
they noticed that the resistances did not return at once to their original values on
cooling. They also observed a considerable increase of resistance in all the coils,
which was most rapid at the outset, and which they attributed to the effect of
recovery from the soldering. We have seen that No. 4086 underwent a similar
increase of resistance at the outset. -Messrs. Harker and Chappuis do not seem to
have observed the long-period effect of hysteresis, which is more insidious. The
correction for this cannot be accurately applied, but it appears that it might explain
some of their errors of observation.

In order to reconcile the observations of Barnes and Fraser with each other
and with those of Glazebrook, it is necessary to suppose that the value of the
temperature coefficient '000254 assumed by Barnes for the McGill Pt-Ag standard
ohms is a little too small. According to my own experiments with these particular
standards, in 1893 and 1894, made by heating the coils in a water-bath, the value
should be ‘000275. I tested only two of the coils, but it is clear from the agreement
of the relative differences observed by Barnes and Fraser that the temperature
coefficients of all the coils are nearly the same. The changes in the relative
differences observed by them are in no case greater than can be explained by the
uncertainty of temperature, as the coils were not immersed in a water-bath in their
comparisons.

Adopting the value ‘000275 for the temperature coefficient of the 10 Pt-Ag
standard ohms, we find the following corrected results in place of (5) and (6),

Barnes, No. 4086 at 21°‘74 C. = 1-00048 ohms .... (8).

Fraser ,, „ „ 14°‘8 C. = 1 ‘00035 ohms .... (9).

The difference between these two values gives a temperature coefficient for No. 4086
of ‘000020, which agrees with their other results supposing that Fraser’s obser¬
vations were taken in the basement within a few days of those of Barnes. In
comparing these results with those of Glazebrook, it must be remembered that
owing to the solidity of the construction and the perfection of the heating arrange¬
ments in the Physics Building at McGill College, the temperature of the Heat
Laboratory very rarely fell below 15° in winter (excejff in a few corners near the
windows), or rose above 22° in summer. No. 4086 was never exposed at Montreal to
so great changes of temperature as it experienced in the temporary observing room
at Kew. The results of the comparisons as regards the absolute values of the

CONTINUOUS ELECTRIC CALORIMETRY.

79

current-standards at McGill College are, therefore, deserving of greater weight than
might be supposed at first sight.

Taking the mean of the observations of Barnes and F baser as fairly applying to
the mean temperature of the Heat Laboratory at the time, we obtain

No. 4086 at 180,4 C. = 1 '00042 ohms in terms of ten McGill standards . (10).

Taking the long-period coefficient '000047 as appropriate for deducing the value
of No. 4086 at the same temperature, from Glazebrook’s observations we find

No. 4086 at 18°'4 C. = 1 '00046 ohms in terms of original B.A. standards (11).

This agreement is as close as could reasonably be expected, and we may conclude
that the value of No. 4086 is known from the comparisons to at least 1 part
in 20,000. Adopting the latter value for 4086, we obtain finally for the value of the
mica current-standards,

Mica current-standards in parallel at 18°'36 C. = '50037 ohm . . (12).

Assuming the temperature coefficient '000248 for the mica current-standards, we
find for the values at 5°, 10°, 15°, and 20°, respectively,

Values of Mica Current-standards in Parallel.

Temp. Cent . 5° 10° 15° 20°

Value . '49871 '49933 '49995 '50057

These values are in practical agreement with those obtained by Barnes from the
McGill comparisons alone, but he does not make any attempt to explain the apparent
discrepancies. They are probably correct in terms of the International Standard
Ohm to 1 part in 20,000, but the uncertainty of the absolute value of the
International Ohm itself may amount to 2 or 3 parts in 10,000. If the Board
of Trade Ohm really exceeds the C.G.S. Ohm by nearly 3 parts in 10,000, according
to the result of the measurements of Professors Jones and Ayrton with the McGill
College apparatus, the electrical watts supplied in the calorimeter have been over¬
estimated to that extent, so that the absolute value of the equivalent of the calorie
in terms of electrical energy as deduced from this investigation would require to
be reduced by about 1 part in 4,000.

It should be observed that the absolute values of the current-standards are of
comparatively little importance for the present investigation, as they do not at all
affect the question of the variation of the specific heat. I have thought it worth
while, however, to discuss the question somewhat fully for two reasons. In the first
place, the comparisons illustrate a possible objection to the use of manganin coils for

80

PROFESSOR HUGH L. CALLENDAR ON

accurate work, which is not I think sufficiently appreciated. In the second place, it
is possible that at some future time the absolute values of the electrical standards
may be more accurately determined, in which case it is desirable that the apparent
discrepancies in the comparisons of the standards employed in the present investiga¬
tion should be clearly explained, in order that the correction may be suitably applied.

The above explanation of the discrepancies observed in the comparisons of the
manganin standard No. 4086, was submitted to Mr. Glazebrook for his approval in
January, 1901. He considered, however, that the measurements in his possession
at that time were not conclusive evidence of hysteresis, as four other manganin coils
of German make, which had been tested at the same time, had shown similar though
not identical variations. He was himself inclined to explain these irregularities by
uncertainties of temperature or of temperature-coefficient of the Pt-Ag standards,
especially at the higher temperature in the July observations. The large value of
the coefficient ‘000047 for the manganin coil, deduced from the July and December
observations, might be explained by supposing that the value of the coefficient
assumed for the Pt-Ag coils was too large. This, however, would not explain the
smaller values of the coefficient for the mana;anin coil deduced from the observations
from day to day, in which the same value of the Pt-Ag coefficient was assumed.

In order to settle the question, a further series of comparisons of No. 4086 was
made in June and July, 1901, at the National Physical Laboratory, which com¬
pletely confirmed the observations made in the previous July, showing that they
were not merely accidental results due to erratic change of No. 4086 or to uncertainty
in the temperature conditions prevailing at that time in the observing room.

As a further test of the possibility of hysteresis theory already advanced, a special
comparison was made, in which No. 4086 was heated through a small range of
temperature and allowed to cool slowly, being tested against a manganin standard
kept at a steady temperature throughout the comparison. This test gave a very
small coefficient, but the curve representing the observations gave unmistakable
indications of hysteresis. It was confirmed by a second test, in which No. 4086 was
cooled, and allowed to warm up slowly. The curve representing the observations in
this case had the opposite curvature. The two curves combined gave a figure
similar to one of the familiar cycles of magnetic hysteresis. These effects could not
be explained as due to a real lag of temperature, since the rate of change of tempe¬
rature was very slow, and the case was filled with oil in direct contact with the wire
and also with the thin metal tube on which the wire was wound. With this direct
evidence of the existence of hysteresis over short periods and small ranges of
temperature, it becomes easier to admit the possibility of larger variations of long
period, such as those indicated by the July and December comparisons with the
platinum- silver standards.

My assistant, Mr. Eumorfopoulos, is now engaged in testing the conditions of
existence of the long-period effect, by a differential method designed for the purpose.

CONTINUOUS ELECTRIC CALORIMETRY.

81

(C.) Current.

(10.) The Electrodynamometer.

For the absolute measurement of the E.M.F. of the Clark cells employed in this
investigation in terms of the standard ohm, I proposed to employ an electro¬
dynamometer of the pattern constructed by Latimer Clark for the Electrical
Committee of the British Association. This instrument is described and figured in
Maxwell’s “ Electricity and Magnetism,” vol. 2, p. 339. The moment of the
couple due to the current is measured by means of a bifilar suspension, the constant
of which is determined by observations of the time of oscillation. It has been
usual of recent years to prefer the current-weighing method, in which the force due
to the current is directly balanced by weights, to methods depending on the
observation of an angular deflection ; but it appeared to me that the British
Association apparatus, with certain modifications of detail, would escape most of the
errors generally urged against the deflection method, and that it possessed important
advantages in other respects.

The electro-dynamometer in the possession of McGill College, as originally
constructed by Messrs. Nalder, was intended to be an exact duplicate of the
apparatus in the Cavendish Laboratory at Cambridge. The coils of the Cambridge
apparatus were wound under the supervision of Maxwell, whose measurements of
the large coils were assumed by Lord Rayleigh in his determination of the
electrochemical equivalent of silver. As a preliminary step, the apparatus was set
up and tested in October, 1894, with the assistance of Mr. R. 0. King, then a
fourth-year student. The defects of the electrodynamometer, in its original form,
were found to be so serious that it was necessary to re-wind the coils, and make
other extensive alterations. This involved considerable delay, while the necessary
materials were being obtained and preparations made ; and the apparatus was
not set up in its final form until the return of Mr. King from Harvard in
September, 1897.

The principal defects of the original apparatus were found to be as follows : —
(1) The framework of the large coils, and the pulley arrangement for equalising the
tensions of the bifilar suspension were not sufficiently rigid. It was not possible
to obtain the desired order of accuracy in the observation of the times of oscillation,
or in the deflections. (2) The most serious defect wTas the uncertainty of the
insulation between adjacent layers of the windings of the coils. This could not be
directly tested, but was inferred from a comparison of the mean radii of the coils
determined by electrical methods with the values calculated from the measurements.
The difference could only be attributed to defective insulation, as the number of
turns and the measurements were verified by unwinding the coils. The error
amounted to nearly 1 part in 500 for the large coils (324 turns each), and about

YOL CXCIX. — A.

M

82

PROFESSOR HUGH L. CALLENDAR ON

1 in 300 for the small coils (576 turns each). The insulation between the coils and
the metal frame-work was in all cases perfect, as the channels were lined with
paraffined paper. The wire itself proved to be very carefully wTound, and the
double silk covering throughout was uninjured. The defects could not he attributed
to excessive damp, as the winter climate of Montreal is extremely dry, and the
laboratory was most efficiently heated. Having regard to the perfect condition of
the wire, it seemed unlikely that absolute security of insulation could be attained
with silk-covered wire, however carefully wound.

Owing to the extreme importance of securing perfect insulation, and of being able
to test the insulation at any time, the small pair of coils were immediately re-wound
with a double winding of two wires side by side throughout, with paper between
each layer. They were then boiled in paraffin wax, and the insulation has since
proved to be practically perfect. At the same time it was decided to re-wind the
large coils with a double-winding of copper tape, of the same width as the channel,
insulated with paraffined paper or silk between the layers. But as there was some
delay in securing the requisite material, the experiment had to be postponed for the
time. I took advantage of this delay to work out the details of the following
improvements, which were introduced into the method on resuming the work in
October, 1897, with Mr. King’s assistance.

(11.) Duplex Scale Reading.

For greater accuracy in observing the deflections and measuring the scale distance,
which is generally one of the weakest points in a deflection method, a duplex system
of reading was adopted. The apparatus was erected on a suitable pier, which had
been provided for it in a basement room, with copper fittings and heaters, where the
temperature was very steady and the ground free from vibration. A plane parallel
mirror, ^ inch thick, and 2 inches in diameter, silvered on both faces, was fitted to
the suspended coils. A pair of metre scales, accurately divided on plane milk-glass,
were mounted with suitable adjustments on a rigid frame of copper tubes at a distance
of 3 metres apart, 150 centims. east and west of the mirror respectively. A
circulation of water was maintained through the copper tubes. The distance
between the scales and the thickness of the mirror could be measured with great
accuracy. The deflections were observed simultaneously from either side with a pair
of very perfect reading telescopes of 2 inches aperture, and about 2 feet long.
These were fitted with filar micrometers, and adjusted so that one turn of the
micrometer screw wTas very nearly equal to 1 millim. on the scale. The mirror
and telescopes were specially made by Brashear for this work. The coils and
suspension wTere completely enclosed to screen them from draughts, and observations
were taken through a pair of thin mica windows. With this arrangement, it was
optically possible to read with certainty to a fiftieth of a millimetre on a steady

CONTINUOUS ELECTRIC CALORIMETRY.

83

deflection of 50 centims. on either side of the zero. This method of reading
eliminates a number of small uncertainties, and makes it possible to approach an
order of accuracy of 1 part in 100,000 in the deflection measurements.

(12.) The Bifilar Suspension.

In order to minimise the effect of imperfect elasticity, the wires of the suspension
were made of hard drawn copper of high conductivity, as fine as was consistent with
constancy of length (safety factor 5), and were rigidly . clamped at a distance of
3 "2 centims. apart, after the tensions had been adjusted to equality. The length
of the suspension was 80 centims., and the effect of torsion of the wires was
comparatively small. The directive force of the bifilar depended almost entirely on
gravity, and remained constant to 1 in 20,000 for several months. The effect of
current heating of the suspension was tested by observing the time of oscillation
with the working current of '5 ampere flowing in opposite directions in the two
windings of the small coils, and was found to be less than 1 part in 20,000.

To determine the moment of inertia, the small coils were fitted with a co-axial
brass tube about 2*5 centims. in diameter, and 50 centims. long, containing a pair of
cylindrical inertia weights, 150 grammes each, which could be clamped at the centre
of the tube, or at the ends. The distance between the weights in their extreme
position is the most important measurement. This was determined by a very
accurate pair of steel callipers by Brown and Sharpe, reading to ‘01 millim., with
a range of 50 centims. This gave sufficient accuracy for the preliminary measure¬
ments, but it was intended to employ a Whitworth measuring machine for the final
series. In observing the oscillations, the times of passage were recorded on an electric
chronograph with a standard clock rated from the observatory. The periods o
oscillation, with the ends of the weights flush with the ends of the tube, and with
the weights in contact at the centre of the tube, were 11 '5385 and 6 '7857 seconds
respectively, and the observations on different days, when corrected for temperature,
did not vary by more than two or three in the last figure.

(13.) Mean Radius of the Large Coils.

The mean radius of the pair of large coils was determined from the length of the
copper tape with which they were wound. This method is not satisfactory with soft
annealed copper wire owing to stretching, but the hard rolled copper tape could be
wound without any tension, and did not undergo any change of length. This was
verified by graduating the tape itself on a 50-foot comparator, the errors of which
were known ; then winding the coil for trial, and unwinding and measuring the tape
again, which was found not to have changed in length by more than a tenth of
a millimetre in each 50 feet. The tape was supported horizontally on the polished
surface of the comparator, and measured under a tension of 6 kilogs. Young’s

m 2

84

PROFESSOR HUGH L. CALLEXDAR OX

modulus was determined for each section. The tape was wound on each coil in two
lengths of 19 turns each, starting at opposite ends of a diameter, with two
thicknesses of paraffined paper between each turn, so that the insulation could be
tested with absolute certainty at any time. There were 38 complete turns, and
nearly 6000 centims. of tape, on each coil. The probable error of the measurement
was less than 1 millim. on the whole length, i.e., less than 1 part in 100,000. The
coils could not be boiled in paraffin after winding, as this would subject the tape to
uncertain strains owing to the contraction of the wax. It was found necessary to
re-wind the coils two or three times with minor improvements before the insulation
proved to be perfect. Finally, silk ribbon was adopted in place of paper.

(14.) Distance betiveen Mean Planes of Large Coils.

The coils were fitted to the frame at a distance between their mean planes equal to
the mean radius of either coil. For this purpose, the original frame, which was
insufficiently rigid, was strengthened with heavy rings of brass turned to fit the
coils, and carefully tested for non -magnetic quality. The distance between the mean
planes, though four times less important than the mean radius, must be known
with considerable accuracy. This was effected by making the coils reversible and
measuring their thickness. Each coil when in position rested against the points ol
three screws, which were adjusted to the right distance apart, and clamped in position.
The distances between the points of opposite screws were measured with the callipers,
and also the thickness of the coils themselves at the points of contact. The coils
were connected to their respective terminals by eight pairs of flexible conductors,
twisted together in pairs so as accurately to compensate, and arranged so that the
coils could be rapidly removed, reversed and replaced at any moment. This appears
to be the only satisfactory method of determining the distance between the mean
planes, as the winding of the coils and the distribution of the current in the tape
cannot be assumed to be perfectly symmetrical with respect to the channels.

(15.) Area of the Small Coils.

The most difficult part of the work was the determination of the area of the small
coils by comparison with the large coils. A thick brass tube, carrying a delicately
suspended magnet and mirror at its centre, was rigidly fixed to the framework so as
to be co-axial with the large coils. The small coils could be mounted co -axially
on this tube at a mean distance of 30 centims. on either side of the centre. Currents
were passed in opposite directions through the large coils and small coils in
parallel, and the resistances of the circuits were adjusted until the magneto¬
meter showed no deflection. The ratio of the resistances was then immediately
determined in a manner similar to that employed by Lord Rayleigh in measuring

CONTINUOUS ELECTRIC CALORIMETRY.

85

the mean radius of the small coil in his current balance. The chief difficulties were
due to change of resistance of the coils with change of temperature. These changes
were considerably reduced by employing copper resistances for the adjustment, and
by arranging the details so as to secure the greatest rapidity of observation.
Readings were taken in each position with the large and small coils severally
reversed, and interchanged, and replaced, the complete series each night including
36 independent readings of the ratio. The distance between the two positions of the
small coils on either side of the centre is the most important measurement. This
was determined by two shoulders turned on the thick brass tube, the distance
between which was measured with the callipers. It was necessary to make all these
comparisons at night between the hours of 2 and 5 a.m. in order to avoid disturbance
from the electrical railway, two lines of which passed within about a quarter of a mile
of the building. The values of the ratio obtained from observations on three different
nights, when there was no magnetic disturbance and the conditions were otherwise
satisfactory, showed extreme differences, amounting to nearly 1 part in 5,000. These
may have been partly due to temperature, as the small coils had been boiled in
paraffin, which must have affected the expansion.

(16.) Ratio of the Currents in the Coils.

In the usual method of employing the electrodynamometer, the same current is
passed through the large coils and small coils in series. This method has the
advantage of simplicity, but it is essential, in order to obtain steady deflections and
secure the maximum accuracy of reading, that there should be no appreciable heating
of the small coils by the current. The latter condition, however, cannot be satisfied
if the currents are equal, unless the large coils are wound with very fine wire, which
is for many reasons objectionable. In the present case, the windings were designed
to secure approximately equal current heating when the currents are in the ratio of
1 to 10 in the small and large coils respectively. The ratio of the currents was
adjusted to this value at each observation by having a standard ohm in series with
the small coils, and a standard tenth-ohm in series with the large coils.
The whole arrangement formed a Wheatstone bridge, the balance of which was
adjusted by means of some copper coils and a high-resistance shunt in series with
the small coils. The standard coils were of manganin immersed in oil, and the
ratio remained extremely constant, owing to the equality and smallness of the current
heating, and the perfection of the insulation. Thus, although the division of the
current involved an additional adjustment of the ratio of the resistances, no
appreciable error was thereby introduced, and great advantages were secured by
the equable distribution of the heat developed, and the steadiness of the observed
deflections.

86

PROFESSOR HUGH L. CALLENDAR ON

(17.) Electromotive Force of the Clark Cells.

The electromotive force of the Clark cells was determined by simultaneous
observations with the potentiometer of the difference of potential on either of the
standard manganin resistances while the currents were being measured by the
electrodynamometer. This observation depended to some extent on the accuracy
of calibration of the potentiometer, but this was repeatedly tested with results
consistent to 1 part in 50,000. Owing to the great steadiness of the current,
there being no silver voltameter in circuit, the potentiometer readings could
easily be taken to this order of accuracy. The Clark cells were of the hermetically
sealed type described in the 4 Proceedings of the Royal Society’ for October, 1897,
and were kept in a well-stirred water-bath at a constant temperature.

The results of the first series of observations on the absolute value of the E.M.F. oi
the Clark cell in terms of the ohm, taken with this apparatus during the spring and
summer of 1898, were sent in by Mr. R. O. King as the report of his third year’s
work as 1851 Exhibition Scholar. When certain minor corrections are applied for
the values of the resistance coils and for the higher harmonics in the series repre¬
senting the force between the fixed and suspended coils, the result found for the
E.M.F. of the Clark cell at 15° C is,

Clark cell = P4334 volts,

the volt being defined as the potential difference due to a current of one-tenth of a
C.G.S. unit through a standard ohm. This result was not immediately published,
because Mr. King hoped to be able to secure an order of agreement higher than
1 part in 10,000 in the comparison of the small coils with the large coils, which was
much the most difficult part of the measurements. But it may be questioned
whether a higher order of accuracy could reasonably be expected in this determi¬
nation, and although the value above given was not at the time intended to be final,
I feel that it may be regarded with considerable confidence on account of the high
order of accuracy of the individual measurements, and the many new devices which
were introduced into the design of the apparatus.

The work of setting up the apparatus and taking the observations was performed
almost exclusively by Mr. King under my supervision, with occasional assistance
from other students in taking readings. I was able to assist him personally during
the vacations in some of the more important measurements, such as the graduation
of the copper tape on the comparator, the first winding of the large coils, and the
first comparison of the small coils. I also verified the accuracy of most of the
adjustments, and the perfection of the insulation, and am satisfied that the whole
work was most carefully and systematically carried out. Mr. King has since left
McGill College, but I hope that he may yet find time to work out and publish in
detail the final results of his observations, which should form a valuable contribution

CONTINUOUS ELECTRIC CALORIMETRY.

87

to absolute electrical measurement. It should be noticed that the result above
deduced from the first series of observations by Mr. King agrees with that deduced
from the observations with the electrical calorimeter by comparison with Howland
and with Reynolds and Moorby, namely, 1 '4332, within the limits of probable error
of the several methods.^

Part III. — Thermometry.

(18.) The Compensated Resistance Box.

Nearly all the temperature measurements in this investigation were made with a
special form of resistance box, which contains some devices which have not as yet-
been adopted in any other instrument of its class, or described in any scientific
periodical. The most important feature in its construction was the system adopted
for compensating the resistances to eliminate the effect of change of temperature. In
the usual form of box the temperature of the coils is taken by means of a mercury
thermometer, and the correction for change of temperature applied from a knowledge
of the temperature-coefficient. The principal objection to this method is that the
mercury thermometer cannot follow the temperature changes of the coils with
sufficient exactness, and that the temperature is generally far from uniform through¬
out the box. In the method of compensation which I patented in 1887 (Complete
Specification No. 14,509) the temperature correction is automatically eliminated by
combining with each of the resistance coils proper, which are made of platinum-
silver or some other alloy possessing a small temperature-coefficient, a compensating
coil of copper or platinum having a large coefficient. The resistance of the compen¬
sating coil is adjusted so that its change of resistance per degree is equal to that of
the coil it is intended to compensate, while its actual resistance at any temperature
is much smaller, the ratio of the resistances being inversely as the temperature-
coefficients. Each compensator is placed in the box in close proximity to the coil it
is intended to compensate, so as to be always at the same temperature, but coil and
compensator are connected on opposite sides of the bridge-wire, so that the balance
depends only on their difference, which remains constant for any change of tempera¬
ture, provided that the adjustment has been properly effected. The advantage of the
method lies in the fact that this adjustment can be made with extreme accuracy, by
testing coil and compensator together over the required range of temperature before
they are connected in their places in the box. But the method has not come into
general use, partly on account of the labour involved in the adjustment of the coils,
and also in part owing to the discovery shortly afterwards of manganin and other
alloys of small temperature-coefficient, which are fairly satisfactory for ordinary work
though inferior, in my opinion, to the compensation method for work of precision
The objections to manganin, for instance, are — (1) That it cannot be perfectly

* The details of the construction and comparison of the Clark and cadmium cells employed are
sufficiently described in the ‘ Proc. Roy. Soc.,’ 1897, vol. 62, p. 117, and by Dr. Barnes, Section 3a, p. 159.

88

PROFESSOR HUGH L. CALLEXDAR OX

annealed after winding, and that its resistance is consequently liable to change for
some time. (2) That the temperature-coefficients of different specimens are often
different, and vary for different sizes according to the method of annealing, so that it
is desirable to test the temperature-coefficient of each coil in the box, and to apply
the corrections separately in the most accurate work. (3) That the temperature
correction cannot be satisfactorily applied, as there is generally some hysteresis in the
change of resistance with temperature, and the values of the resistances depend to
some extent on previously existing conditions of temperature. (4) That it cannot he
hard soldered without burning, and that soft solder connections to manganin are
frequently found to be defective, unless they have been most carefully made.
(5) That it is liable to corrosion if exposed to damp or gas fumes, although the usual
coating of shellac is sufficient protection in most cases. These defects have been
noted by other observers, notably by Harker and Chappuis, who employed a box
with manganin coils in their recent comparison of the platinum and gas-thermometers
The majority of other alloys of this class are inferior to manganin in constancy; they
are also frequently objectionable (e.p., “ constantan ”) on account of their great
thermoelectric power, which produces inconvenient disturbances if the temperature ot
the box is not uniform. The advantage of platinum-silver lies in the perfection with
which it may be annealed, and in the absence of lag or change when properly
annealed.

If uncompensated coils of platinum-silver are used, it is necessary to keep them in
an oil-bath to secure sufficient certainty of temperature. It is also desirable to
employ a thermostat for regulating the temperature of the oil-bath, and a stirrer for
keeping the temperature uniform throughout. This adds considerably to the cumber¬
someness and expense of the apparatus, and to the difficulty of using it. It is really
simpler in the end to use compensated coils, as the individual temperature-coefficients
of the coils must otherwise be determined and corrected separately, at least for work
of the highest accuracy.

The first apparatus constructed on this principle was made in 1887, shortly before
applying for the patent, and was figured in the specification. All the parts of this
apparatus were made interchangeable in pairs, by an extension of the Carey-Foster
principle, with a view to facilitate testing and calibration. There w,ere two exactly
similar bridge-wires, each a metre long, and three pairs of compensated coils, which
could be inserted singly or in series, with resistances on the binary scale equal to 1,2,
and 4 times that of either bridge-wire. The platinum-thermometers were also made
in pairs, after the pattern described and figured in the ‘ Phil. Trans.,’ A, 1890, but
were generally contained in separate tubes, which greatly facilitated construction,
and permitted them to be used differentially by a simple change in the connections of
the bridge. In the instrument used for the boiling-point of sulphur ( loc . cit .) it was
necessary to have both thermometers in the same tube. This construction was also
adopted in 1887 for comparison coils in which it was necessary that both wires should

CONTINUOUS ELECTRIC CALORIMETRY.

89

be accurately at the same mean temperature. The necessary changes of connection
were effected by means of mercury cups and thick copper connectors, similar to those
employed for standard resistance coils. The mercury cups were made by boring
suitable holes in a flat plate of ebonite, to the under side of which thick copper plates
were screwed, the joint being made mercury tight with a thin sheet of rubber.

The apparatus above referred to was first used for some determinations of the
linear expansion of standard yards at the Standards’ Office in 1887. It was subse¬
quently employed by Dr. A. S. Lea, F.R.S., and Dr. W. H. Gaskell, F.R.S., and
later by Dr. Rolleston, in some physiological experiments on the heat produced in
muscle and nerve by electrical and other stimuli. For this purpose a very delicate
pair of differential thermometers were constructed of ‘001 -inch wire wound on mica,
weighing a few milligrammes each, and sensitive to the ten-thousandth of a degree C.
One of these thermometers was described and figured in the specification (No. 14,509,
1887). This apparatus is still in the possession of the Physiological Laboratory,
Cambridge.

The chief defect of the original form of apparatus was the uncertainty of the
temperature correction of the platinum-silver bridge-wire owing to its length. In
making a new form of apparatus in October, 1890, bridge- wires of manganin
were employed, annealed at a red heat in coal gas. The pair of bridge-wires could be
very accurately calibrated throughout their length by the Carey-Foster method, but
owing to the trouble of determining and applying the bridge-wire correction, it was
eventually decided to use a bridge- wire of low resistance in conjunction with a larger
number of resistance coils. It also proved to be unnecessary in practice to make all
the coils interchangeable in pairs, provided that the ratio coils were tested for equality
of temperature-coefficient. In this case, it was sufficient to calibrate the bridge-wire
and resistances by a method of substitution, which was much simpler than the
Carey-Foster method. Apparatus constructed on this principle was described in the
‘ Phil. Mag.,’ July, 1891, and figured in the patent specification (No. 5342, 1891). As
the resistance coils were no longer required to be interchanged, they were permanently
connected to the copper plates in a single box instead of being connected to copper
rods in separate boxes like standard coils. The mercury cups, however, were still
retained, in preference to plugs for short circuiting the resistance coils in accurate
work, and were constructed precisely as originally described. The simplification
consisted in connecting the resistance coils permanently in series, and using simple
bridges of thick copper connected in pairs for short-circuiting each resistance coil
and its compensator simultaneously.

The particular resistance box employed in this investigation is shown in the
accompanying fig. 2. It was made to my designs by the Instrument Company,
Cambridge, at the beginning of 1893, but I had personally to undertake the delicate
work of compensating and adjusting the resistance coils. It contained 9 resistance
coils, A, B, C, D, &c., on the binary scale, ranging from 25 GO to 10 units, constructed

VOL. CXCIX. — A.

N

90

PROFESSOR HUGH L. CALLENDAR OK

of platinum-silver annealed at a red heat, and compensated with similarly annealed
platinum. The box measured 15” X 6'5" X 3‘ 5". The bridge-wire was of platinum-
silver, nearly 34 centims. long, hut only 5 centims. on either side of the middle
were actually used in the measurements. The scale was of brass, divided by
Troughton and Simms to half millimetres, with a vernier reading to a hundredth of
a millimetre. To secure this order of accuracy in the readings, the contact piece,
consisting of a short length of the same platinum -silver wire with a nearly sharp
edge, was rigidly fixed to the sliding piece carrying the vernier. The bridge-wire
was stretched at such a height as to clear the contact edge by about '01 inch, and
contact was effected by pressing down the bridge-wire on to the contact wire by
means of an india-rubber finger. This finger was provided with a screw adjustment,
so that the contact could be set and held at any desired point. To keep the tension

& 0 0

| w uy " |

Fig. 2. Compensated Resistance Box.

of the bridge-wire constant, which is most important in accurate work, the wire was
stretched between parallel bars of brass and iron in an intermediate position corre¬
sponding approximately to its coefficient of expansion. Connection was made to the
galvanometer through a similar wire stretched parallel to the bridge-wire, in order to
eliminate thermal effects at the sliding contact. All the connecting wires throughout
the box were accurately paired and compensated, and the thermometer connections
were made by means of mercury cups at PP' CCk Screw terminals were used in the
galvanometer and battery circuits only, where changes of resistance are immaterial.
The ratio coils were adjusted to equality by the method of interchanging, and were
tested and compensated for equality of temperature- coefficient before being fixed in
their places in the box. They were not wound up together, as is usual in apparatus
of this class, but were merely fixed side by side, as it is most important to secure the
most perfect insulation of the ratio coils for delicate differential work.

CONTINUOUS ELECTRIC CALORIMETRY.

91

The box was originally intended for working chiefly with thermometers of pure
platinum having a zero resistance equal to the largest coil, i. e. , 2560 box-units, and a
fundamental interval of 1000 box-units, or nearly 10 ohms. This gave a very
convenient scale of 10 centims. of the bridge-wire to the degree, and made it possible
to take readings to the ten-thousandth part of a degree. Thermometers of double
the resistance, having a scale of 20 centims. to the degree, were employed for some
differential work, e.g., for demonstrating the lowering of the melting point of ice due
to one atmosphere of pressure ('0075° C.) at the May Soiree of the Royal Society in
1893, where this box was first exhibited. But for ordinary temperature measure¬
ments, unless the insulation were extremely perfect, it was found that very little
could be gained in point of accuracy by going beyond 10 ohms for the fundamental
interval. For the boiling-point of sulphur, a fundamental interval of 5 ohms was
found to be preferable ; and 1 or 2 ohms for higher temperatures where the insulation
was necessarily less perfect and the conditions less steady. As the portion of the
bridge-wire actually utilised never exceeded 5 centims., and averaged only 2 ’5 centims.,
or \ of a degree, it was seldom necessary to take any account of the calibration
correction of the bridge-wire, the errors of which proved to be less than one part in
500, or one two-thousandth part of a degree, without correction. In the best
mercury thermometers it is unusual to calibrate closer than 2° intervals, and the
corrections are necessarily uncertain to two or three thousandths, even if the inter¬
polation formulae can be trusted to one part in a thousand, which is very doubtful. It
is easy to see how great is the advantage of the platinum -thermometer in point of
ease and accuracy of calibration.

The comparison of the 9 coils of the resistance box, which were arranged on the
binary scale, could be carried out and the relative values calculated in less than an
hour. The peculiar advantages of the binary scale for this purpose are frequently
misunderstood or misrepresented. The most important of these advantages are :
(1) That a given resistance can be represented only by one particular combination of
coils, so that there can never be any doubt as to which combination was employed
for any given reading ; (2) That the least possible number of separate coils are
required, and that the complete calibration requires the least possible number ot
readings, and can be effected in the least possible time. It is often urged as an
advantage of less simple and symmetrical arrangements, that each resistance can be
made up in a great variety of ways, which act as checks in case of doubt ; and that
the relative values of the coils can be compared in a number of different combinations,
so that several equations can be obtained for evaluating each resistance. This is, no
doubt, an advantage in cases' where the accuracy of calibration depends on micrometric
estimation, as in calibrating a mercury-thermometer. But, if a good galvanometer is
available, the accuracy of comparison of resistances is not limited in this way. We
may, therefore, fairly consider the disadvantage of the excessive expenditure of time
in the observations and calculations ; moreover, the risk of error due to changes of

N 2

92

PROFESSOR HUGH L. CALLENDAR ON

condition or temperature, which is the really important point to be considered in
work of this kind, is increased by the greater complexity and want of symmetry of
the system of comparison. Examples of the calibration of this box on the binary
method are given by Barnes, p. 187.

(19.) Heating of the Thermometers by the Measuring Current.

It is generally assumed in the construction and use of apparatus for comparing
standard resistance coils, that the conditions of greatest sensitiveness are obtained
when the resistance of the ratio arms is equal to that of the resistances to be
compared. This, however, is certainly not the case in platinum-thermometry, and
seldom in other cases, unless the heating of the resistances by the measuring current
can be safely neglected. In platinum-thermometry the heating of the thermometer
by the current is the limiting consideration which determines the amount of power
available for the test. For a given rise of temperature with a given thermometer
the current must not exceed a certain value, e.g ., about a hundredth of an ampere
for a rise of temperature of a hundredth of a degree, with an average platinum-
thermometer of •006// wire (‘Phil. Trans.,’ A, 1887, p. 184). With this limiting
condition it is easy to see (1) that if the ratio coils are in parallel with the
thermometer, the sensitiveness is doubled by making the ratio coils very small ;
(2) that if the ratio coils are in series with the thermometer, the sensitiveness is
doubled by making the ratio coils as large as possible. In every case it is necessary,
in order to secure accurate compensation for the variation of the leads, that the ratio
coils should be equal, and that the bridge-wire should be inserted in the circuit
between the thermometer and the adjustable balancing coils. The arrangement (2)
with the ratio coils in series with the thermometer, involves leading the battery
current in through the bridge-wire sliding contact, which is generally unadvisable on
account of possible disturbance produced by variation of resistance at the contact, or
by breaking the circuit to readjust. The first arrangement is therefore generally
adopted, and some advantage is gained in this case by making the resistance of the
ratio coils considerably smaller than that of the thermometers, provided that they
are not made so small as to be appreciably heated when the working current is
passed through the thermometer. In this particular box the ratio coils had a
resistance of 6 ’4 ohms, or about a quarter of the normal thermometer at 0° C.
They were each constructed of two '008" platinum-silver wires in parallel, and
adjusted for equality of temperature-coefficient. With a working current of '002
ampere through a 2 5 '6 ohm. thermometer constructed of '004" wire, the heating
effect in the thermometer would be nearly a thousandth of a degree, and the current
through the ratio coils would be nearly one-hundredth of an ampere, which would
not produce any material rise of temperature.

CONTINUOUS ELECTRIC CALORIMETRY.

93

In differential work it is seldom necessary to take any account of current heating
of the thermometers, unless the difference of temperature is considerable, or the
thermometers are very differently situated. With a single thermometer, it is
desirable to measure the heating effect occasionally, especially if a galvanometer of
suitable sensibility is not available, or an excessive current is employed for any other
reason. The simplest method of determining the rise of temperature due to the
current in any case is to use two similar cells of low resistance, preferably storage
cells, which can be connected in series or parallel by changing a switch. The normal
measurements are effected with the cells in parallel. On putting the cells in series,
the current through the thermometer is very nearly doubled, and the heating effect
is nearly quadrupled, provided that it is small. The correction for current heating is
obtained by subtracting from the first reading one-third of the difference between the
two readings. I have used this method in all accurate work for the last ten years,
and it appears to be worth recording, as there is some conflict of opinion with regard
to the proper method of procedure. Harker and Chappuis measured the heating
effect of the current on one of their thermometers at 0° C., and, assuming that the
effect would vary as the watts expended on the coil, they adjusted the external
resistance in the battery circuit so as to give always the same ivatts in the coil at
different temperatures. This is not quite correct, since the cooling effect of
conduction and convection-currents of air in the tube increases nearly in proportion
to the absolute temperature. The effect of radiation also becomes important at high
temperatures, and the cooling is then more rapid. If, therefore, the watts are kept
constant, the heating effect will diminish as the temperature rises, and a small
systematic error will be produced. Assuming that the rate of cooling increases as

t

the absolute temperature 9, and that the watts are kept constant, the heating effect
at any temperature 9 is 273 h/9, where h is the heating effect in degrees of
temperature at 0° C. It is easy to see that the corresponding systematic error in the
temperature t on the centigrade scale, would be approximately ht(t — 100)/373(U-|-273).
In the case described by Harker and Chappuis (‘ Phil. Trans.,’ A, 1900, p. 62), the
heating at 0° C. was ,014° C. The systematic error at 50° C. would be only ‘0003°,
and at 445° C. only '008°.

A better rule is to keep the current through the thermometer constant. In this
case the heating effect is nearly constant, since the resistance of the thermometer
increases very nearly as fast as the rate of cooling, i.e., a little faster than the
absolute temperature. In this case it is evident that the error would be negligible,
even if the heating effect at 0° C. were as large as a hundredth of a degree. In any
case, provided that a galvanometer of suitable sensibility is used, the error due to the
heating effect will be practically negligible, even if no account is taken of it, i.e., if
the resistance in the external battery circuit is kept constant. It is assumed, of
course, that the current is kept flowing through the thermometer continuously, so
that the heating effect is steady. Some - writers have advised keeping the circuit

94

PROFESSOR HUGH L. CALLENDAR ON

closed for the shortest possible time. This method should never be used in accurate
work. Other writers have apparently found more serious difficulties, and appear to
have considered that the heating effect was fatal to accurate work. This view has
arisen merely from the use of unsuitable apparatus or faulty arrangements.

(20.) Ice-point Apparatus.

In accurate work the heating effect should never exceed a hundredth of a degree,
and the correction can be readily applied if required, in case the current through the
thermometer is not kept constant. It is possible, however, to obtain consistent
results, even if the heating effect amounts to several tenths of a degree, provided

that the conditions are steady, and that the heat gener¬
ated is not allowed to accumulate. This condition is
generally satisfied in a bath of saturated steam or vapour,
or in a well-stirred bath of liquid, but not at the freezing
or melting point of a bad conductor. Errors due to
variation of the heating effect are most common in
observing the ice-point. The density of the water at
this point is nearly constant, so that the convection
currents are feeble, and the thermometer, if the current
is excessive, or if there is considerable conduction of heat
along the stem, as in the case of thick porcelain tubes,
may become surrounded by a layer of water at a
temperature appreciably above the freezing-point. Some
advantage is gained in this case by employing a stirrer
to make the water circulate vigorously through the ice.
For this purpose I devised the following apparatus,
which proved very useful for investigating the heating
effects of large currents at the freezing-point, where
accurate results could not be obtained by merely sur¬
rounding the thermometer with ice. The apparatus
consists of two concentric cylindrical vessels. The ther¬
mometers are placed in the inner vessel. The whole
apparatus is filled with melting ice, with enough water
from previously melted ice to fill up the interspaces.
The bottom and top of the inner vessel are fitted with
gauze strainers to prevent circulation of the ice. The
water is caused to circulate through the ice by means of a centrifugal stirrer below
the middle of the inner vessel, worked by a shaft passing up through the centre of
the apparatus. The thermometers should be deeply immersed, if they have thick
rubes, or copper or silver leads, in order to minimise the effect of conduction along

Fig. 3. Ice-point Apparatus.

A, gauze-covered apertures ;
B, outer vessel ; C, inner
vessel ; D, shaft of stirrer ;
E, gauze strainer; F, lagg¬
ing ; G, thermometer; H, ice ;
J, stirrer blades.

CONTINUOUS ELECTRIC CALORIMETRY.

95

the stem. It is probable that a similar method would give the most accurate
results in all .cases, but if the current heating is small its variation may generally
be neglected, so that no special apparatus is required.

(21.) Insulation of Thermometers.

Defective insulation due to moisture condensed in the tubes is sometimes a source
of error in accurate work at the ice-point with thermometers of high resistance,
if the tubes are not sealed. To avoid this, the instruments may be fitted with a
small inner tube leading to the bottom, through which dry air may be forced
occasionally. A better plan, which I first adopted in 1893, for accurate work at
low temperatures, is to seal the platinum leads through the glass so that the whole
thermometer is air-tight. In this case the platinum leads may conveniently
terminate in glass cups, and may be connected to the external leads by mercury or by
fusible alloy,* as indicated in fig. 4. If the tubes are made of lead-glass, there is no

a

Fig. 4. Hermetically-sealed Thermometers.

B, brass tube ; C, compensator ; D, plaster of Paris ; Cf, glass tube ; M, mica discs ; P, pyrometer leads.

difficulty in fusing the four leads through the tube. The joint will even stand
sudden exposure to high-pressure steam without cracking, if properly annealed. All
the thermometers employed in my experiments on the temperatures of steam in the
cylinder of a steam-engine (‘Proc. Inst. C.E.,’ November, 1897), were made in this
manner in order to make the joint perfectly steam-tight at high pressures. The
method cannot be applied to porcelain tube pyrometers, but in this case the employ¬
ment of a high resistance is out of the question for other reasons. I recommended
hermetic sealing for the Kew thermometers (‘ Nature,’ October, 1895), but it was
not adopted, as the instruments were of low resistance, intended primarily for high
temperature work. With such an instrument it is easy, as the recent report of
the Kew Observatory shows (‘ Proc. Poy. Soc.,’ November, 1900), to obtain an order
of accuracy of a hundredth of a degree on the fundamental interval, which is all
that is required for work at high temperatures ; but it would be unreasonable to
expect to be able to work to the thousandth of a degree, except under the best
conditions, with the most perfect apparatus and the most skilful observers.

* I originally employed common solder, but the fusible alloy, first suggested by Griffiths, appears to
make a good connection and is more easily managed, though it could not be used in the steam-engine
experiments.

96

PROFESSOR HUGH L. CALLENDAR ON

(22.) Differential Measurements.

The only thermometric measurement requiring the highest accuracy in the present
investigation was the difference of temperature between the inflowing and outflowing
water at the two ends of the calorimeter. As this difference was obtained by a
single reading of a pair of differential thermometers, all the minor errors and
corrections were practically eliminated. Further, as there was no particular
advantage to be gained by observing the fundamental interval with greater pro¬
portionate accuracy than the differential measurement, many special precautions,
such as measuring the current heating, or keeping the current constant, were
rendered superfluous. For the same reason it was unnecessary to secure the highest
possible accuracy in the comparison of the coils of the box (as only a few’ of the
smallest were used for the differential measurement), and no attempt was made to
keep the temperature of the box approximately constant (to minimise possible errors
due to imperfect compensation, or variation of temperature), although such pre¬
cautions would naturally be adopted in many other investigations. It was sufficient
to make sure that the vernier was correctly set and read, and that the sensitiveness
of the galvanometer was suitable for the differential measurement.

The galvanometer employed in this investigation was specially made for the work
by Messrs. Nalder Bros., to my order, in 1896. It was an astatic instrument,
with four small coils of 5 ohms each. It had a fairly small and light magnet system,
and a light plane mirror about 4 millims. in diameter, and was fitted with a long
silk suspension, and a symmetrical system of control magnets above and below.
The suspension might have been more delicate, and the magnet and mirror system
lighter, with a smaller moment of inertia, but as the sensitiveness proved to be
ample for the purpose, no alterations were made. The most important point for our
work was the astaticism of the magnets. It was quite impossible to work with a
sensitive non-astatic galvanometer owing to disturbance from the electric railway.
But the astatic instruments, if properly adjusted, were very little affected. It was
necessary to keep the time of swing short to secure sufficient rapidity of observation,
but even with this restriction there was no difficulty in adjusting the galvanometer
to give a deflection of four or five scale-divisions for a thousandth of a degree with a
current of about one two-hundredth of an ampere through the thermometers. The
deflections were observed with a microscope carrying a fine scale and a micrometer
eye-piece. No attempt was made to adjust the contact to the exact balance point.
It was merely set to the nearest millimetre division of the bridge-wire, and the
galvanometer deflection observed on reversing the battery current. The exact
balance reading could be easily calculated from this by observing the deflection per
millimetre of the bridge-wire, which remained fairly constant. The mean temperature
difference was worked out to the ten-thousandth of a degree. It appears probable
from the observations that the error very rarely amounted to as much as one two-
thousandth of 1° on a rise of temperature of 8°.

CONTINUOUS ELECTRIC CALORIMETRY.

97

(23.) Reduction of Results to the Hydrogen Scale.

A most important factor in the variation of the specific heat of water, as Rowland
was the first to point out, is the correction to the absolute scale of the readings of the
particular thermometer employed in the research. In the present case, the tempera¬
ture observations were taken with platinum-thermometers of standard wire, and were
approximately reduced to the absolute scale by the difference -formula,

t ■— pt = dt (t : — 100)/10,000 . (l).

The recent observations of Messrs. Harker and Chappuis (‘Phil. Trans.,’ A,
1900) with a constant-volume nitrogen thermometer have confirmed my conclusion
that a formula of this type represents the deviation of the platinum-thermometer
from the absolute scale within the limits of error of observation over the range 0°
to 600° C. A similar conclusion follows from the comparison by Griffiths (‘ Phil.
Trans.,’ 1893) and by Waidner and Mallory (‘ Phil. Mag.,’ July, 1899) of mercury-
thermometers standardized at the International Bureau with platinum-thermometers
from 0° to 25°, and from 0° to 50°.

The value of the difference-coefficient d in this formula was assumed to be I *50
from the mean of a number of observations taken at different times with different
samples of the wire. The thermometers themselves were not directly tested, as the
variation of the difference-coefficient for different specimens of the pure wire is in
nearly all cases less than the probable error of a single determination. The
particular sample of wire employed for thermometers E, on which most of the
results depend, was tested by Mr. Tory (‘ Phil. Mag.,’ 1900) by comparison with the
original standard wire from which it was drawn, and found to be identical.

The limits of variation of the difference-coefficient for pure platinum wire, as
tested by competent observers, assuming the normal boiling-point of sulphur to be
4440,53 C., on the scale of the constant-pressure air-thermometer, is only P49 to
1*51. The greater part of this variation is probably due to errors of observation
and differences of annealing. A variation of (ROl in the value of d would affect
the specific heat of water by only 1 part in 10,000 at 0° or at 100° C., and by much
less at intermediate points of the range. If we had assumed the boiling-point of
sulphur to be 4450,27 C.,# as found by Harker and Chappuis with a constant-
volume nitrogen-thermometer, the value of d would have to be increased to 1*54,
which would increase the values of the specific heat of water by only 4 parts in
10,000 at 0° C., and by 2 in 10,000 at 25° C. The difference in the above values of
the boiling-point of sulphur may possibly be explained as due to a real difference in

* Chappuis has recently (‘Phil. Mag./ 1902) accepted the results of LIolborn and Day for the
expansion of Berlin porcelain, which reduce his value for the boiling-point of sulphur to 444° '7 C. — Added
March 11, 1902.

VOL. CXCIX. — A.

O

98 PROFESSOR HUGH L. CALLENDAR ON

the scales of the constant- volume and constant-pressure thermometers, as ve have little
direct experimental knowledge of the relation of the scales at these temperatures.
It may be equally due to systematic differences of reduction of the observations,
as for instance in the application of the correction for the expansion of the
envelope of the gas-thermometer. The correction applied by Chappuis,* obtained by
extrapolation of a formula deduced from observations over the range 0° to 100° C.,
is much larger than similar corrections found by Callendar and Bedford for hard
glass and porcelain from observations over the range 0° to 600° C. The lower value
of the boiling-point of sulphur obtained by Callendar and Griffiths in 1890
was confirmed by Callendar with a different instrument in 1893, and later by
Eumorfopoulos. Since the uncertainty of the correction for the expansion of the
envelope is so great (Callendar, ‘ Phil. Mag.,’ December, 1899), it was decided to
adopt the older value, which has been in use for 10 years, rather than to attempt
a special correction based on the probable scale difference of the constant-pressure
and constant-volume thermometers.

The uncertainty of this correction is mainly due to the difficulties of gas
thermometry. The scale of a platinum-thermometer constructed of pure wire is
so easily and so accurately reproducible, that it appears practically certain, as I have
already explained at some length in a previous paper (‘ Phil. Mag.,’ December, 1899),
that it would afford a more convenient standard of reference than the hydrogen
thermometer for scientific purposes. By employing a standard difference-formula,
such as (1) for reduction of platinum temperatures to the absolute scale, we should
obtain results in sufficiently close agreement with the thermodynamical scale for all
practical purposes, and we should be saved the trouble and confusion incidental to
small uncertain corrections. From this point of view it would be more scientific to
omit any further reduction to the hydrogen scale, but it may be of interest to
indicate the nature and the probable magnitude of the correction.

Since the parabolic difference-formula ( 1 ) was established and verified by means of
observations with air- or nitrogen-thermometers, it would be most natural to assume
that the scale obtained by its application coincided very closely with that of the
nitrogen-thermometer, and to reduce the results to the hydrogen scale by the
application of the table of corrections given by Chappuis, deduced from the
following formula for the difference,

tn -th — t{t- 100) (+ 6-318 + -00889* - -001323*2) X 10~6 . . (2).

This formula has been applied to our results by Griffiths (‘ Thermal Measurement
of Energy,’ Cambridge, 1901), who gives a table of the corrected values. But there

* Chappuis has recently (‘Phil. Mag.,’ 1902) accepted the results of Holborn and Day for the
expansion of Berlin porcelain, which reduce his value for the boiling-point of sulphur to 444° "7 C. — Added
March 11, 1902.

CONTINUOUS ELECTRIC CALORIMETRY.

99

are some objections to be considered. Chappuis’ formula (2) refers to the constant-
volume nitrogen-thermometer at 100 centims. of mercury initial pressure, whereas
the difference-formula (l) was obtained with a constant -pressure air-thermometer at
76 centims. pressure. Moreover, formula (2) makes the difference negative between
73° and 100°, as shown in the second column of Table IV., so that the correction to
the specific heat would change from —2 in 10,000 at 80° to +6 in 10,000 at 100°.
The negative differences are of the same order as the probable errors of the
observations. Chappuis himself considered them to be impossible, and gave a
revised formula for the mean coefficient of expansion of nitrogen, from which the
“corrected” values in the third column have been calculated. He has since recal¬
culated the values on a slightly different assumption, namely that the pressure-
coefficient- ( dp/dt)/p0 of nitrogen at an initial pressure p0 = 100 centims., reaches a
minimum value '0036738 at 80° C., and then remains constant at all higher tempera¬
tures. Taking the fundamental coefficient (0° — 100°) as being '00367466, the
difference of the scales above 100° C. would be linear, and would amount to '023° per
100°. The effect of this assumption between 0° and 100° does not differ materially
from M. Chappuis’ first “ corrected ” results.

It is interesting to compare Chappuis’ results with those calculated from the
observations of Joule and Thomson. In order to represent the results of these
observers more accurately, especially in the case of hydrogen, I have added a term b to
their formula, to represent the “ co-volume,” as in the later equations of Hirn and
Van der Waals. The equation of Joule and Thomson then becomes

v-b = R 6/p - A/6'2, . (3)

which is practically equivalent for moderate pressures to the formula devised by
Clausius to represent the divergences of C03 from Van der Waals’ formula. The
differences calculated from this formula for nitrogen and hydrogen at constant-volume
and 100 centims. initial pressure, are of the same order of magnitude, but not quite
so large, as the “ corrected ” differences of Chappuis. On the whole the agreement
appears very satisfactory. It would have been still closer if the nitrogen-differences
observed by Chappuis in the second series of his observations between 0° and 25°
had not been raised by '007° in order to make the curve pass through the zero point.
(See Callendar, ‘ Proc. Phys. Soc.,’ March, 1902, where the subject is more fully
discussed.) It should be remarked, on the other hand, that the observations of
Joule and Thomson can be represented well within the limits of experimental error
by the formula of Van der Waals. According to the latter formula the pressure at
constant- volume is a linear function of the temperature, and the differences between
the scales of all constant-volume thermometers should be identically zero. The
evidence of the experiments of Joule and Thomson taken alone is therefore incon¬
clusive, but it may be stated that the observations of Amagat, Witkowski, and

o 2

100

PROFESSOR HUGH L. CALLENDAR OX

others on the compressibility of gases over considerable ranges of temperature and
pressure, indicate a real difference between the scales, similar to that calculated from
the observations of Joule and Thomson by the modified equation (3).

The differences between the nitrogen and hydrogen scales at a constant pressure of
1 atmosphere calculated from the observations of Joule and Thomson by the same
equation are given in the same table. In each case the difference between the
hydrogen scale, tjt, and the absolute scale, 6, is added. In the case of the constant-
pressure thermometer the correction is larger than at constant-volume, but there is
less uncertainty in its value, as the results calculated by different formulae ( e.g ., Van
der Waals and Clausius) are very nearly the same.

Table IV. — Difference between Scales of Nitrogen and Hydrogen Gas-

Thermometers.

Temperature,

Centigrade.

Chappuis,
formula (2).

Chappuis,

corrected.

Joule-Thomson,
constant- volume, 100 centims.

Joule-Thomson,
constant-pressure, 7 6 centims.

in ~~ ih-

in ih •

ih - 0*

tn-ih.

ih -

o

10

+ -0057

+ -0053

+ -0030

+ • 0005

+ -0062

+ -oon

20

+ -0095

+ -0087

+ '0050

+ -0009

+ -0103

+ -0019

30

+ -0113

+ -0105

+ -0065

+ -0012

+ -0131

+ -0024

40

+ -0110

+ -0110

+ -0073

+ -0013

+ -0143

+ -0026

50

+ -0086

+ -0103

+ -0073

+ -0013

+ -0144

+ -0026

60

+ -0049

+ -0090

+ -0066

+ -0013

+ -0130

+ • 0025

70

+ -0010

+ -0069

+ -0055

+ -0011

+ -0112

+ -0020

80

- -0023

+ -0045

+ -0044

+ -0008

+ -0083

+ -0015

90

- -0032

+ -0022

+ -0025

+ -0004

+ -0044

+ -0008

In order to reduce the value of the specific heat of water expressed in terms of any
scale of temperature t! to the corresponding value expressed in terms of any other
scale of temperature t", it is only necessary to multiply by the factor dt'jdt". This
factor is readily obtained if the formula giving the relation between t' and t " is
known. For instance, in order to reduce from the platinum scale by means of the
difference-formula, we obtain at once by differentiation of the formula the factor

dpt/dt = 1 + (100 - 2t) cZ/10,000 . (4).

Since the specific heat of water varies so little from unity, the correction to be
added at any point may be taken as being practically equal to the excess of the
correction factor above unity. The corrections from the nitrogen to the hydrogen
scale, obtained by differentiation from the table of differences above, are very uncer-

CONTINUOUS ELECTRIC CALORIMETRY.

101

tain, and are for this reason given only to the nearest part in 10,000. The platinum-
scale correction is given for comparison in the last line of the table.

Table V. — Corrections (Parts in 10,000) to be added to reduce the Value of the
Specific Heat of Water from the Nitrogen to the Hydrogen Scale.

Temperature, Centigrade ....

0°

10°

20°

30°

40°

50°

o

o

o

o

so

80°

90°

100°

Chappuis, formula (2) .

+ 6

+ 5

+ 3

+ 1

-1

-3

-4 -4

- 2

+ 1

+ 6

,, corrected .

+ 6

+ 4

+ 3

+ 1

+ 0

- 1

-2 -2

-2

— 2

- 2

Joule-Thomson, constant-volume,

100 centims .

+ 4

+ 3

+ 2

+ 1

+ 0

-0

- 1 - 1

-2

-2

-3

J oule-Thomson, constant-pressure,
76 centims. . .

+ 7

+ 5

+ 3

' +2

+ 1

-1

-2 -2

-3

-4

-5

Platinum scale correction by differ-

ence-formula (1) .

+ 150

+ 120

+ 90

+ 60

+ 30

0

-30 -60

-90

-120

- 150

It will be observed that the correction in any case is very small, and that the
uncertainty of the correction is nearly as large as the correction itself. A change of
nearly 1 in 1000 in the correction at 100° is produced, if we adopt the “ corrected”
results of Chappuis instead of the table taken by Griffiths. On the whole, as the
difference-formula (l) was obtained by comparison with a constant-pressure air-
thermometer at 76 centims., we shall probably be most nearly correct if we apply the
corrections calculated from the observations of Joule and Thomson for air under the
same conditions. The corrections thus obtained do not differ materially from those
applied by Griffiths (they agree to 1 in 5000), except at 90° and 100°, where they
differ by 5 and 11 in 10,000 respectively. I have therefore assumed the corrections
calculated from Joule and Thomson in reducing the results to the hydrogen scale in
Table XII., Section 37.

The value 1'54 for the difference-coefficient already referred to was obtained by
Harker and Chappuis with a constant- volume nitrogen-thermometer at 56 centims.
initial pressure. The corrections for this case can be calculated from the observations
of Joule and Thomson by reducing those for the constant-volume thermometer given
in Table IV. in proportion to the initial pressure, namely, in the ratio 56 to 100. It
happens that if our results are reduced by employing Chappuis and Harker’s value
of the difference-coefficient, 1*54, and are then corrected to the hydrogen scale by
applying the correction for the constant- volume nitrogen-thermometer at 56 centims.,
the results are identical with those obtained by using the difference-coefficient, 1*50,
and then applying the correction for the constant-pressure air-thermometer. This
agreement, however, is not really so satisfactory as it appears at first sight, because,
according to the theory on which it is based, the correction to the hydrogen scale
does not follow the same function as the difference-formula, and the difference in the

102

PROFESSOR HUGH L. CALLENDAR ON

values of the coefficients assumed, namely, 1‘50 and 1'54, cannot be entirety explained
by the difference of the scales at the boiling-point of sulphur. #

Part IV. — Calorimetry.

(24.) Temperature Regulation.

The question of temperature regulation was particularly important in this method
of calorimetry. It also presented exceptional difficulties on account of the form of
the calorimeter, and the large range of temperature to he covered. The apparatus
enqiloyed for this purpose was made in the summer of 1895, and I spent a good deal
of my leisure time during the session 1895-96 experimenting with various forms of
regulator before I was able to obtain a satisfactory arrangement.

On account of the extreme length of the calorimeter (over 1 metre), and since it

Fig. 5. Heater, Circulator, and Regulator.

AB, oil bulb ; C, stirrer blades ; D, gland ; E, inflow ; F, F, outflow.

was necessary to have both the ends accessible for inserting leads and thermometers,
I decided to employ a tube form of water-jacket open at both ends in place of the
more usual bath and stirrer. The most convenient method of maintaining- such
a jacket at a constant temperature appeared to be by means of a vigorous water
circulation maintained by a centrifugal pump. The apparatus constructed for this
purpose is shown diagrammatically in fig. 5. It was intended to serve as heater,
regulator and circulating pump simultaneously. The annular bulb AB was filled
with oil, the expansion of which actuated a gas regulator in the usual manner. The
revolving blades CC of the centrifugal pump were connected through the gland D
with a water motor by means of a short piece of stiff rubber tubing. The circulating

* The Joule-Thomson equation, extrapolated to 445°, would make the constant-volume thermometer
read two or three-tenths of a degree higher than the constant-pressure thermometer, which would account
for the difference between Chappuis’ corrected result, 444°-7 C., for the sulphur boiling-point, and the
value, 444°-53 C., obtained by Callendar and Griffiths. — Added March 11, 1902.

CONTINUOUS ELECTRIC CALORIMETRY.

103

water was sucked in through the central tube E, 1 inch in diameter, and ejected
through the annular space surrounding the bulb of the regulator by the tube F.
The whole was mounted on a large gas burner, and shielded with an asbestos screen.

The object of placing the regulator bulb inside the heater, and in close proximity to
the pump, was to secure quickness of action in response to any change in the gas-
pressure. The violent stirring, and the comparatively small capacity of the heater in
proportion to the bulb favoured this result. It was most important that the regulator
should be very sensitive, and that there should be no forced oscillations of temperature,
because the jacket-temperature determined that of the inflowing water-supply to the
calorimeter, and did not merely affect the external heat-loss. Any temperature
oscillation would j3roduce a serious effect on the results, especially on the smaller
flows, in which the total mass of water passing, about 250 grammes in 15 minutes,
was not very large compared with the effective thermal capacity of the calorimeter,
which was about 50 grammes.

When the regulator was made sufficiently sensitive to cut off the gas for a very
small change of temperature, I found it necessary at the higher points of the range,
where a large supply of gas was required, to adjust the by-pass so that, if the
regulator were cut off, the temperature would very nearly reach the required point.
Forced oscillations could only be avoided if the regulator controlled a very small
fraction of the heat supply, acting merely as a fine adjustment on the temperature of
the system. Under these conditions, however, when the total gas-supply was large,
any small accidental change in the gas-pressure might exceed the limits of control of
the regulator. A rapid change in the quality of the gas produced similar effects. It
was therefore absolutely necessary at the higher temperatures to keep the gas-
pressure very constant. The best forms of gas-governor were tried, but did not
prove sufficiently delicate. I therefore fitted up a large copper gas-holder, delicately
suspended and counterpoised by means of a steel tape passing over a wheel with ball
bearings so as to move with very little friction. This arrangement proved to be
capable of regulating the pressure to within a tenth of a millimetre of water.
The large capacity of the gas-holder tended also to minimize the effect of sudden
small variations of quality of the gas, such as might be produced by air in the
pipes, &c.

The action of this constant-pressure gas-supply was so perfect that for many
purposes no other temperature-regulator was required, and as a matter of fact none
was used in many of the preliminary experiments. Since variations of gas-pressure
were practically eliminated, it was found to be unnecessary to have the regulator bulb
inside the heater, and the fine adjustment-regulator was placed in the tank C,
(Barnes, figs. 14 and 15, pp. 211, 213). This regulator was employed to operate an
electric heating arrangement, as described by Dr. Barnes in Section 4, in which slow
period oscillations were prevented by the device of the reciprocating contact suggested
by Gouy (‘ Journ. de Phys.,’ 1897, p. 479). I found it necessary to introduce a few

104

PROFESSOR HUGH U CALLENDAR ON

modifications into the arrangement as described by Gouy, the most important of which
was to make the relay put another lamp in series, instead of breaking the circuit.
This prevented the destructive sparking at the break, which was a serious matter
with a 100 candle-power lamp. Of course, the greater part of the work still fell on
the constant-pressure gas-supply, which was mainly responsible for the excellence of
the results at high temperatures.

(25.) Preliminary Experiments on the Specific Heat of Mercury.

Our first experiments by the steady-flow electric method of calorimetry were made
on the specific heat of mercury, as it presented fewer experimental difficulties than
water. Since mercury is itself a conductor of suitable specific resistance, it was
unnecessary to insert a heating conductor in the fine flow-tube, which greatly
simplified the fitting together of the apparatus. The conducting properties ol
mercury were also utilised in the design of an electrical device for maintaining
a constant head so as to secure uniformity of flow. The level of the mercury in the
reservoir was regulated by a platinum wire contact which actuated an electro-magnet

Fig. 6. Diagram of Mercury Calorimeter.

compressing a small rubber tube which supplied mercury to the reservoir. As soon
as the level fell below the platinum point, the contact was broken, the armature
released, and more mercury supplied. The inflow was arranged to keep the mercury
near the platinum point in perpetual agitation, so that there was no sticking or
hunting of the regulator. The whole arrangement was fitted to a wooden bracket
belonging to a Geissler pump, which could be hung up on the wall at different levels
when it was required to alter the flow of mercury in a given ratio. The flow was
regulated and steadied by passing the mercury through fine glass tubes immersed in
a tank of water before entering the calorimeter. In measuring the flow of mercury,
the time of switching over was automatically recorded on an electric chronograph by
the momentary contact of the mercury thread with an edge of platinum foil, which
diverted the flow from one beaker to another.

The design of the calorimeter itself will be readily understood from the accom¬
panying diagram (fig. 6). The inflow and outflow tubes AB and CD are exactly
similar, about 2 centims. internal diameter and 25 centims. long. They are
connected by the fine flow-tube BC, of 1 millim. bore and 1 metre long, coiled in

CONTINUOUS ELECTRIC CALORIMETRY,

105

a short spiral of about 2 ‘5 centims. diameter. The vacuum jacket EF extends about
13 centims. along the inflow and outflow tubes, and is provided with a side tube for
exhausting. The inflow and outflow tubes are provided with two side openings, the
smaller of which was intended for the inflow or outflow of mercury, and the larger
for the leads conveying the electric current. In some of the earlier experiments the
larger tubes were fitted with a pair of delicate mercury-thermometers for watching
the progress of the experiment, and observing when the conditions became steady.
But these mercury-thermometers were found to be of little or no use, and the large
side-tubes were subsequently removed to facilitate the fitting of the calorimeter in
a tubular form of water-jacket.

(26.) Method of Determining the True Mean Temperature of Outflow.

By far the most important point in this method of calorimetry is the device
adopted for obtaining the true mean temperature of the outflowing liquid, and
securing a definite measurement of the electrical watts expended in heating it. If
a thermometer were merely inserted in the outflow-tube, leaving a free space all
round for the circulation of the liquid, it is evident that the heated liquid would tend
to flow in a stream along the top of the outflow-tube, and that the thermometer
might indicate a temperature which had little or no relation to the mean temperature
of the stream. It is easy to make an error of 20 per cent, in this manner, as I
found in my preliminary experiments in the summer of 1896. A fairly uniform
distribution of the flow might he secured by making the space between the
thermometer and the outflow-tube very narrow. But this leads to another difficulty
in the case of mercury. As the space is narrowed, the electrical resistance is
increased, and an appreciable quantity of heat, which cannot be accurately estimated,
is generated in the vicinity of the thermometers.

Both the difficulties above mentioned were overcome in the mercury experiment
by fitting the inflow and outflow tubes with soft iron cylinders, 6 centims. long,
turned to fit the tubes, and bored to fit the thermometers. The soft iron had a
conductivity about ten times that of mercury for both heat and electricity. The
heat generated by the current in the immediate vicinity of the thermometer bulbs
was so small that the watts might fairly be calculated from the difference of
potential between the iron blocks at the middle points of the bulbs. The mercury
stream was forced to circulate in a spiral screw thread of suitable dimensions cut in
the outer surface of the blocks, which prevented the formation of stream-lines along
one side of the tube, and secured uniformity of temperature throughout the cross
section of the outflow-tube. The high conductivity of the iron also assisted in
securing the same result.

A precisely analogous device for averaging the outflow temperature was applied in
the water calorimeter. The bulb of the thermometer was fitted with a copper sleeve

VOL. CXCIX. — A.

P

106

PROFESSOR HUGH L. CALLENDAR OX

of high conductivity, on the outside of which a rubber spiral was wound to fit the
outflow-tube as closely as possible. The accuracy of fit was found to be much more
important in the case of water than in the case of mercury. The reason of this is
that, the thermal conductivity of water being 10 or 15 times less than that of
mercury, the accurate averaging of the outflow temperature is more dependent on
the uniformity of the spiral circulation and the complete elimination of asymmetric
stream-lines.

In order to obtain a perfect fit for the sleeves with their spiral screws, it was
necessary that the bore of the outflow-tube should be as nearly uniform as possible,
and accurately straight. It was most essential that there should be no constriction
at the points of junction E and F with the vacuum -jacket, and that the external
portions of the tubes AE, FD should not be of smaller bore than the portions inside
the vacuum-jacket, though it would not matter much if they were a little larger.
These details of the design, which determined the choice of the dimensions of the
tubes, had all been carefully worked out before the ordering of the first six
calorimeters with vacuum-jackets in October, 1896, and the importance of the
straightness and uniformity of the tubes was clearly explained in the specification.
There was some difficulty in making the calorimeters accurately to specification, and
when they arrived about three months later, it was found that this particular detail
had been somewhat overlooked. It was consequently a difficult matter, even with a
soft rubber spiral, to secure sufficient perfection of fit, and the accuracy of many of
the earlier experiments was seriously impaired.

The effect of an imperfect fit was to permit part of the heated stream to escape
directly past the thermometer, so that the temperature indicated by the outflow
thermometer was lower than the true mean of the flow. This defect was less
apparent with a large flow or a large rise of temperature, either of which conditions
tend to promote mixing of the liquid and the attainment of a proportionately greater
uniformity of temperature. A good illustration of this is afforded by the results
quoted by Dr. Barnes in Section 7 of his paper, p. 237, which were obtained with one
of the first three calorimeters in which the bore was undoubtedly defective. The
apparent diminution of the heat-loss per degree rise with increase in the rise of
temperature is probably due in part to the more perfect mixing of the stream caused
by the greater differences of temperature, which promote instability of flow and
increase the formation of eddies. I may add that I have repeatedly observed the
same effect in my experiments on steam by a similar method. With steam it is
much more difficult than with water to secure a true average of the outflow
temperature. Any imperfection of fit or circulation immediately produces the
observed effect in an exaggerated form.

The greater irregularity of the results (Barnes, Section 7) for the small flows is
probably in part accidental, but is also characteristic of imperfect fitting of the
rubber spiral in the outflow-tube, which is obviously more detrimental in the case of

CONTINUOUS ELECTRIC CALORIMETRY.

107

the smaller flows. As a consequence of this irregularity of the small flows. I do not
think these results can be fairly treated by the somewhat artificial method adopted
by Dr. Barnes. The extrapolation of the heat-loss per degree for zero rise of
temperature is too uncertain in the case of the small flow. The natural method of
treatment would be to take the difference between the sum of the electrical watts
SEC, and the sum of the heat-watts SJQ (16, and divide by the sum of the
temperature differences Scl9, to find D for each flow. Combining the two flows in the
usual manner, without any arbitrary assumptions, we thus obtain practically the
same result as that deduced by Dr. Barnes.

(27.) Design of the Water Colorimeter.

The design of the water calorimeter presented certain points of difficulty which
were not settled without some preliminary experiments on the conditions of flow in
fine tubes. The greater part of these experiments were carried out in the summer of
1 896, in the Thermodynamical Laboratory of the Engineering Building of McGill
College, with the kind permission of Professor Nicolson, who erected for this purpose
a large supply-tank of water on an upper floor in a room at a very constant
temperature. The results of some of these experiments were mentioned in a paper on
the ‘ Law of Condensation of Steam,’ which was communicated to the Institution of
Civil Engineers in September, 1896, and read in the following year.

Since water is practically a non-conductor of electricity, it was necessary either to
make the fine flow-tube of platinum instead of glass, or else to thread a conducting
wire or strip of platinum through it. A fine bore metallic flow-tube would have
presented some advantages in point of smallness of radiation loss, but in the end 1
decided to use a glass flow-tube and a central conductor, chiefly on account of the
importance of securing the greatest possible constancy and perfection in the vacuum-
jacket. This could be most easily and certainly attained by making the vacuum-jacket
entirely of glass.

The glass-work of the water calorimeter differed from that of the mercury calori¬
meter (fig. 6) only in having a straight flow-tube 50 centims. long and 2 millims.
bore, instead of a spiral flow-tube 100 centims. long and 1 millim. bore. It was
necessary to make the flow-tube straight on account of the difficulty of threading the
conductor through it as well as the connecting wires to which it was attached. This
operation would have been facilitated by using a larger tube, but, apart from this
necessity, it was desirable to have the flow-tube as fine as possible to secure uniformity
of temperature of cross-section and other advantages. From one point of view, it was
desirable to make it as short as possible, in order to minimise the heat-loss, which
depended chiefly on the surface of the flow-tube; but on the other hand it was
necessary to have sufficient length to eliminate leakage of the current through the
water, and to secure a suitable resistance for the conductor, and sufficient surface to

p 2

108

PROFESSOR HUGH L. CALLENDAR ON

prevent excessive superheating by the current. It was found that the dimensions
above given satisfied the required conditions very fairly, and although there were a
few minor details which could not he satisfactorily settled until the complete apparatus
had been fitted up, it was not found necessary to make any changes in the essential
features of the design. The three calorimeters ordered two years later by Dr. Barnes
were of the same design in all important points, except that two of them had 3 millims.
bore flow-tubes ; but these were found to be less suitable, and were employed in very
few of the tests.

(28.) Improvements in the Design of the Calorimeter.

Although Dr. Barnes was naturally unwilling to introduce any radical changes in
the original pattern, which had proved to be capable of giving very good results, there
can be no doubt that it was capable of improvement, and I had in fact already noted
several points in which alterations were desirable.

The importance of uniformity of bore in the flow-tube, and particularly in the out¬
flow-tube, has already been referred to (§ 26). This was remedied in the later
apparatus. Two of the side tubes were also removed, and the other two bent parallel
to the flow-tube, to facilitate insertion in the cylindrical form of jacket. These
modifications were of small importance, and were made in the first calorimeters after
the apparatus was received. A more important improvement in the same direction
would be to have the tube for exhausting the vacuum-jacket inserted at F, fig. 6, in a
direction parallel to the flow-tube, which would permit the water-jacket to be made
much smaller, thus securing a more vigorous and uniform circulation.

It would be most important for future work to endeavour to reduce (1) the risk of
error from conduction at the outflow end of the calorimeter at high temperatures ;
(2) the correction for the heat-loss, which amounted to 4 per cent, at the higher
points of the range.

The conduction error might be reduced by including a greater length of the outflow-
tube in the vacuum-jacket. This would he much more effective than lagging the
exposed parts with flannel, since the flannel lagging is exposed to the temperature ot
the laboratory, which is much lower than that of the jacket. Besides, the lagging is
apt to become damp, and takes a long time to reach a steady state. The length of
outflow-tube inside the vacuum-jacket might easily be increased to 25 centims. instead
of 12, with 5 centims. outside. This would greatly diminish the possible uncertainty
of the conduction loss.

The heat-loss depends chiefly on the extent of surface of the flow-tube and ther¬
mometer bulb. The thermometer bulb could not be made much smaller, but the
greater part of the loss arises from the flow-tube, which might be reduced to about
half the external diameter, although the bore coidd not conveniently be made less than
1-5 millims. This would also be an advantage as diminishing both the heat capacity

CONTINUOUS ELECTRIC CALORIMETRY.

I 01)

of the tubs and the difference of temperature between the inside and outside of the
glass. In heating the apparatus during exhaustion, there is some risk of breaking the
flowr-tube, as it heats and cools very slowly in a good vacuum. The difference of
temperature between the thick flow-tube and the jacket in some cases caused the tube
to bend so as nearly to touch the sides of the jacket. This effect would be reduced
by making the tube thinner.

It does not appear that very much could be gained by taking excessive precautions
to improve the vacuum. With a less perfect vacuum there would probably be less
variation of the heat-loss, due to the evolution of minute traces of gas during exposure
to a high temperature. The greater certainty of the correction in that case might
compensate for its larger magnitude. The first three calorimeters, one for mercury
and two for water, were exhausted in the laboratory on a five-fall Sprengel pump
which I had set up some time previously for experiments on radiation and on X-ray
tubes. This pump gave a very perfect vacuum, but the tubes were merely heated by
hand with a bunsen burner during the process. They showed, however, approximately
the same rate of heat-loss as the best of the calorimeters which were subsequently
exhausted by Messrs. Eimer and Amend in an asbestos oven. The vacuum in one of
the calorimeters exhausted in the laboratory was also tested by means of a powerful
oscillating discharge from a battery of Leyden jars passed through a coil surrounding
the tube. This failed to induce a ring discharge inside the vacuum-jacket, although
the same test wTould produce a brilliant discharge in the majority of commercial
vacuum vessels for liquid air.

Probably the most effective method of reducing the heat-loss would be to silver the
inside of the vacuum-jacket. I have tried this method in my experiments on the
specific heat of steam by the electrical method with a vacuum-jacket calorimeter, and
have found it very advantageous, owing to the low radiative power of the silver. The
vacuum-jacket in this case requires very careful evacuating on account of the difficulty
of drying the silver film.

With these improvements, the heat-loss could probably be reduced to about one-
quarter of its value in the existing apparatus, and the uncertainty of the correction
would be greatly diminished, though perhaps not quite in the same proportion. In
any case the results obtained with a calorimeter 1 laving a very different value of the
heat-loss could not fail to be a valuable confirmation of the previous work.

(29.) Effect of Variation of Viscosity.

The rise of temperature of the water due to friction in its passage through the
tube, can be easily estimated from the observed difference of head at the inflow
and outflow. 1 made this observation for each of the conductors mentioned on p. 117,
with a flow of half a gramme per second in a tube very slightly less than 2 millims.
diameter and 50 centims. long. The difference of head was found to vary from

1 1 0

PROFESSOR HUGH L. CALLENDAR ON

20 to 30 centims. of water at 20° C., for different arrangements of the conductor, and
different conditions of flow, being greater when the flow was non-linear. Since 1° C.
corresponds to a fall of 42,700 centims. under gravity, the rise of temperature due to
the friction would be less than a thousandth of a degree. This is a quantity which
ought not to he neglected in working to a ten-thousandth of a degree, but the effect
was practically eliminated by the method of observing the difference of temperature,
which was expressly intended to eliminate small residual sources of error of this
character. For each value of the flow, the difference of temperature was observed
“ cold” before turning on the electric current, and the “ cold reading” was subtracted
from the difference observed with the current passing. We are therefore concerned
only with the change in the head due to diminution of viscosity with rise of tem¬
perature when the current is turned on. I found by calculation from the known
variation of the viscosity, and also verified by direct observation in each case, that
this amounted to only 10 per cent, of the head for a rise of 8° C. at 20 C. This
would be equivalent to 2 or 3 centims. fall, or less than a ten-thousandth of a degree,
a quantity which might safely be neglected. The correction would be much smaller
at higher temperatures, owing to the great diminution in the viscosity.

In the final apparatus, as employed by Dr. Barnes, the difference of head would
be somewhat greater owing to the rubber spirals on the copper sleeves, and the
rubber cord on the central conductor. The question was raised at the Dover meeting
of the British Association, and I wrote to Dr. Barnes asking him to measure the
head under the actual conditions of experiment, but the apparatus happened to be
dismounted at the time. We may safely conclude, however, that the difference
of head could not have exceeded 1 metre of water, in which case the correction
would be less than 1 in 40,000 at 20° Cl, and might be fairly neglected. This
correction corresponds with that for the heat generated by stirring in the Griffiths
and Schuster, methods of calorimetry. It amounted in Griffiths’ apparatus to
about 10 per cent, of the heat-supply, but was apparently negligible in Schuster’s
experiments.

(30.) Radial Distribution of Temperature in the Fine Flow- Tube.

We assume in the elementary theory of the experiment that the temperature
is uniform across the section of the flow-tube at any point. It is important to
consider how far and under what conditions this is true. Given the rate of external
heat-loss at any point it is easy to calculate the difference of temperature between the
inside and outside surfaces of the glass tube, but the distribution of temperature in
the liquid can only be calculated if we assume the flow to be in straight lines
parallel to the axis of the tube, and the conductor to be circular in section and
concentric with the tube. Even in this simple case the solution cannot be made
complete, owing to the variations of viscosity and conductivity with temperature ;

CONTINUOUS ELECTRIC CALORIMETRY

111

but it is possible to estimate the order of magnitude of the differences involved,
which is all that is really required for our purpose.

Assuming that the internal and external diameters of the flow-tube are 2 and
G millims. respectively, and length 50 centims., and that the conductivity of the
glass is ‘0020 cal. C.G.S., it is easy to calculate that the mean difference of
temperature between the internal and external surfaces would be of the order of
one-tenth of a degree only when the final rise of temperature is 8°, and the heat loss
•050 watt per degree, as in the majority of our experiments. It will therefore
evidently be unnecessary to take account of this in any of the calculations.

To find the radial distribution of temperature in the liquid, assuming the flow to
be linear, I will take first the case of the metallic flow-tube, which is much the
simplest. The differential equation of the radial distribution of temperature, neglecting
tbe minute effect of longitudinal conduction, is

d (kr dd I dr) I dr — + vcr dd/dx . (1),

in which k is the thermal conductivity of the liquid, and c the specific heat per unit
volume, v the velocity of the stream, d the temperature, r the distance from the axis,
and x the distance along the tube. The velocity v is a function of r, which can be
easily calculated if the viscosity is assumed constant. As a matter of fact, both
the viscosity and the conductivity vary rapidly with change of temperature. The
viscosity at 100° is nearly six times less than at 0° C., and its variation is accurately
known. But if we assume both conductivity and viscosity constant (as we are
practically compelled to do, since the variation of conductivity with temperature is
quite uncertain) we shall obtain a solution which is sufficiently simple to be useful,
and which can be strictly applied to small changes of temperature.

To simplify the solution still further, I shall assume the longitudinal temperature
gradient dd/dx constant over the cross-section of the tube at any point, and equal
to O' /l, where d' is the rise of temperature observed in a length I. This will not
be true near the inflow end of the tube, where the radial distribution of temperature
is rapidly changing, but it will very fairly represent the limiting state, which is
attained when the liquid has flowed along the tube for some distance.

If the flow is linear, and the viscosity constant, the velocity at any point of the
cross-section is given in terms of r by the equation

v

= 2V (1 - (r/r0Y) = 2Q (I - (r/r0)2)/nr0

(2),

where V is the mean velocity, Q the flow in cub. centims. per second, and r0 the
internal radius of the flow-tube.

Making this substitution in (l) and integrating, we obtain

k r dd /dr

06/ / U
7T/ W

(3)-

112

PROFESSOR HUGH L. CALLEXDAR OX

The constant of integration B is determined by the consideration that the tempera¬
ture-gradient vanishes at the centre of the tube. Putting this condition in (3), we
find B = 0, which materially simplifies the solution of the equation. Integrating
from the temperature 60 of the surface of the tube, we find

~ 0 = Q 0' (r*/W - (r/rvf - })/2 nlk . . .

The temperature 0l at the axis of the tube where r = 0, is given by

6U — 6*i = 3Q 6' / Snlk

(4

(5).

The mean temperature of the flow, allowing for variation of velocity over the
cross-section, is given by the expression

60 — 0.-, = llQdy iSnllc

(6).

(31.) Electrical Method of Measuring the Thermal Conductivity of a Liquid.

The remarkable simplicity of this expression induced me to attempt a method of
measuring the conductivity of a liquid, based on the observation of the difference of
temperature 90 — 61, between the tube and the mean of the flow at any point ; the
temperature 0O of the tube and the gradient O' jl being deduced from observations of
the changes of resistance of the flow-tube itself. Although this may appear at first
sight a difficult and out of the way method, it possessed special attractions for me as
an application of the electrical resistance method of measuring temperature, and it
really offers several advantages which more than counterbalance the difficulty of the
electrical measurements. The longitudinal distribution of temperature in the flow-
tube was deduced from observations of the resistance of consecutive sections by the
same method which I had already applied in 1886 to the determination of the
conductivity of platinum. The difficulty of this part of the work was therefore
largely discounted by previous experience. The advantage of the method is that
the tube is its own thermometer ; the temperature measured is that of the tube
itself, and not that of a thermo-couple or water-bath assumed to be at the same
temperature as the tube. This avoids the most common and insidious source of error
in all conductivity measurements.

As compared with the plate-method of measuring the conductivity of liquids, which
has been practised by Weber and many other observers in different forms, the tube-
method possesses several important advantages, (a) It avoids the difficulty of measuring
accurately the small distance between the bounding surfaces of the liquid, or the
thickness of the sheet, since the expressions (5) and (6) already given are independent
of the radius of the tube, and contain only lengths and differences of temperature
which are easily observed, (b) All uncertainties with regard to the area from which
the heat is conducted, and all difficulties of boundary conditions, which cannot

CONTINUOUS ELECTRIC CALORIMETRY.

113

satisfactorily be eliminated in the plate-method, even by the employment of a guard-
ring, are easily avoided in the tube-method by making the tube small in proportion
to its length, (c) The error of the plate-method due to direct radiation through the
liquid, which is quite important with a thin transparent stratum, is completely
eliminated, since all the heat which is lost by the inner surface of the tube must be
absorbed by the liquid itself.

Graetz, (• Wied. Ann./ vol. 18, p. 79) has applied a non-electrical steady-flow
method to the determination of the conductivity of a liquid, which bears a close
superficial resemblance to that above described, but in reality differs from it in several
fundamental points. In his method a stream of liquid at a temperature between
30° and ICbO. flows through a thin capillary tube, 10 centims. long, and 0'6 millim.
bore, immersed in a water-bath at a temperature of 7° C. to 1 0° C. The mean
outflow temperature is observed, and is also calculated in terms of the conductivity on
the assumption that the flow is linear. The temperature of the external surface of
the tube is assumed to be the same as that of the water-bath. This is the most
obvious defect of his method, as the assumption could not be even approximately
true unless the current of liquid through the tube were extremely slow.
Unfortunately in that case the difference of temperature between the outflow
and the bath, on which the measurement depends, tends to vanish ; it is also more
difficult to obtain the true mean temperature of a small stream owing to defective
mixing in the outflow tube, and accidental sources of error due to end-effects are
exaggerated. These defects might be avoided in various ways. The true mean
temperature of the tube itself might be determined by making the tube very thin,
and observing its expansion , or preferably its electrical resistance. Or the tempera¬
ture of the outside surface might be indefinitely approximated to that of the bath by
making tbe tube very thick, and supplying a vigorous circulation around it. The
true mean temperature of the outflow might also be determined for small flows by
adopting a spiral circulation similar to that employed in the present investigation.
There would remain, however, a most essential point of distinction between the two
methods.

In the electrical method, heat is continuously supplied by the current at a nearly
constant rate as the liquid flows along the tube. The observation depends on the
limiting difference of temperature at the end of a long tube when a steady radial
distribution has been reached. The advantage of this is that the solution is
independent of the initial or variable state, the calculation of the effect of which is
much less certain on account of the steep gradients and excessive differences of
temperature involved, which tend to produce disturbances in the flow. In Gkaetz’
method the initial differences, amounting to 20° or 30° Cr, were much larger than in
the electrical method, and the result entirely depends on the correctness cf the
assumptions made in the solution of the initial state. The final or limiting state in
his method is one of uniform temperature, and cannot be utilized at all.

VOL. CXCIX. - A. Q

114

PROFESSOR HUGH L. C ALLEN DAE ON

In carrying: out the electrical method, advantage was taken of the increase of
resistance of the tube with temperature in order to secure a constant temperature
gradient in the latter part of the flow-tube by suitably adjusting the current, as
explained below in § 34. The results were not quite as good and consistent as I had
hoped to obtain, on account of want of uniformity in the platinum-tube employed.
I therefore thought it best to defer publication till I could find time to repeat the
observations under better conditions, but the preliminary work was distinctly
encouraging, and was particularly valuable as an indication of .effects to be expected
in steady -flow electrical calorimetry.

(32.) Superheating of the Central Conductor.

The case of a glass flow-tube with a concentric conductor, which more nearly
approaches the arrangement actually employed in the present investigation, leads to
nearly the same differential equation, but the solution is much less simple. In the
course of designing the experiment, I worked out the complete solution for this case
also, including the initial state, on similar assumptions of constant viscosity and
conductivity. But since the conductor cannot be held exactly central in practice,
and the other theoretical conditions cannot be realized, the method cannot con¬
veniently be applied with a central conductor to the measurement of the conductivity
of liquids. For the purposes of the present investigation, moreover, since we are
only concerned with the approximate estimation of a small correction, a less elaborate
calculation will be more appropriate. In order not to overburden the paper with
purely mathematical difficulties, it will suffice to give the solution of the limiting
state for the simpler case in which the velocity of flow is assumed to be constant over
the cross-section of the tube and equal to its mean value. This simplification does
not materially alter the general character of the solution, and the numerical results
which it gives for the calorimeters actually employed are within a few parts per cent,
of those obtained when allowance is made for the variation of the velocity.

If we integrate the differential equation (l) on the assumption of a constant
velocity Y = Q/(rp — r02)', where r, is the radius of the glass-tube, and r0 the radius
of the conductor, writing A for Yc6'/2lk, we obtain the solution

dd/dr = A r - A r~/r . (7),

00- 6 = Arp log, (r(rQ) - A (r» - r*)/2 . (8),

in which the constant is determined by the condition that, neglecting external heat-
loss, the gradient is zero at the surface of the glass. This gives for the difference of
temperature between the surface of the wire and the surface of the glass,

0o - 0i = aY2 log* (rjrf) - A (rp - ?y)/2

(9).

CONTINUOUS ELECTRIC CALORIMETRY,

I 15

The mean temperature 0.2 of the outflow deduced from (8) is given by

0O ~ = Ar:4 log, (rj/r0)/S — Ar^/2 - AS/4 .... (10),

where S is the sectional area n (rq2 — r02) of the flow, and A = Q0' /27r/IS. Equation (10)
gives the superheating of the conductor above the mean temperature of the outflow.
The difference (9)-(l0) gives the error of the assumption that the glass is at the
same temperature as the mean of the outflow at the outflow point. The limiting
value of the latter error, if r02 is negligible in comparison with rf1 (the most
unfavourable case), is 0.2 — 0X = AS/4 = Q0'/8Trlk.

Although the above formulae cannot be directly applied to the experiments on the
specific heat of water, it is interesting to make an estimate of the superheating of
the conductor, and of the difference of temperature between the glass and the water
under the assumed conditions of linear flow and concentric conductor. It is obvious
from the formulae already given, that the differences of temperature in each case are
directly proportional to the heat supplied by the electric current, and inversely
proportional to the length of the tube and the conductivity of the liquid. To
estimate the effects numerically, we may take the rate of heat supply as
Qff = 5 calories per second, or 21 watts, for the larger flows. The conductivity k of
water at 25° C. may be taken as ‘0016 C.G.S., but is much too uncertain to permit
the estimate to be extended to other temperatures. Since l = 50 centims., we have
inlk = l'OO very nearly.

For the metallic flow- tube from equation (6) the superheating of the tube above
the mean temperature of the flow in the limiting state would be about 5° C., and
would be independent of the diameter. For the glass flow-tube, from equations (9)
and (10), the temperature of the glass would be from 10,5 to 2"-0 below the mean of
the outflow for tubes of the dimensions employed, increasing to 2°‘5 as a limit for a
very large flow-tube with a very small conductor. In spite of its higher temperature
the metallic flow-tube would have the advantage of a smaller heat-loss, owing to its
smaller surface (l millim. diameter instead of 6 millims.), and far lower radiative
power. It would also be possible to measure the actual temperature of the metallic
flow-tube at any time from its resistance, without any knowledge of the conductivity
of the liquid, and without assuming the flow to be linear.

The superheating of the conductor in the glass flow-tube would naturally depend
on the size of the conductor as well as that of the tube, as given by equation (10).
With a wire '8 millim. in diameter, and the flow-tube 2 millims., the superheating
of the wire would be about 4°‘5 for a heat supply of 21 watts. With a wire ‘4 millim.
diameter, and a 3 millims. flow-tube, the superheating would be about 130,2.
This illustrates the importance of having a large surface for the wire and a small
flow-tube. It is probable, however, that the superheating would not directly affect
the radiation loss, as platinum is a bad radiator, and water is very opaque to heat-
rays from heated water.

115

PROFESSOR HUGH L. CALLENDAR OX

(33.) Methods of Eliminating Stream-Line Motion.

It was evident from the equations above given that, since the conditions of linear
flow gave rise to a systematic variation of the temperature of the flow-tube, which
was directly proportional to the flow for a constant rise of temperature, it would be
necessary to adopt some device for mixing the water in its passage through the tube
so as to produce a nearly uniform distribution of temperature over the cross-section
of the tube. The obvious method of securing this result was to employ a stranded
conductor. This would diminish the superheating by increasing the available surface
of the conductor, and would distribute the heat evenly over the cross-section of the
tube, provided that- the strands were separated and arranged in such a manner as to
break up the stream-lines.

In order to verify the theory and observe the nature of the effects to be expected,
I made some rough preliminary experiments on the superheating of various
conductors in a 2 millim. tube with a steady flow of water. The general character of
the flow, and the degree of mixing attained, were observed by the introduction of a
colour-band of blue ink, after the method employed by Osborne Reynolds, and
generally practised in hydraulic laboratories in studying the flow of liquids. The
most instructive results were obtained with a stranded conductor consisting of
5 strands of ,00G" pure platinum wire. My reason for selecting this particular size
of wire was that I happened to possess a considerable quantity of it, and that its
temperature-coefficient was accurately known, as it was regularly employed for
maki ng thermometers.

The resistance of the stranded platinum conductor, when carrying the heating
current, was measured by the Wheatstone-bridge method, by comparison with a
specially constructed platinoid resistance connected in series with it. The two were
connected in parallel with a post-office box, by means of which the ratio of the
resistances was observed. The resistance in the arm connected to the platinoid strip
was 2000 ohms, the resistance in the adjustable arm corresponding to the platinum
conductor was about 6000 ohms. As the platinoid strip remained practically
constant, the resistance of the platinum could be taken as proportional to the
resistance in the adjustable arm. The watts on the conductor were observed by
means of a Weston ammeter and voltmeter of suitable ranges.

The platinoid strip resistance, which was subsequently utilised for regulating the
current, is shown very clearly at L in the bird’s-eye view (Barnes, fig. 15, p. 213). It
consisted of a number of strips of platinoid about 1 foot long, one-half inch broad, and
-fs inch thick, having a resistance of nearly yjjth of an ohm each. The ends of the
strips were bent at right angles and amalgamated. They could be connected in series
or parallel in a great number of different combinations by means of the mercury cups
and copper connectors shown in the plate. For this particular experiment they were

CONTINUOUS ELECTRIC CALORIMETRY.

117

arranged two in parallel, and 19 in series, giving a resistance of nearly 3 th of an
ohm, capable of carrying a current of upwards of GO amperes without excessive
heating. The maximum currents used in this experiment were about 8 amperes.

The rise of temperature of the platinum conductor in each case was found to be
nearly proportional to the watts expended. This was verified by varying the
number of cells employed, and keeping the water-flow constant, for each arrange¬
ment of the conductor to be tested. The superheating of the conductor was
estimated by deducting from the observed rise of temperature half the calculated
rise of temperature of the water, making a suitable allowance for the heat-loss. The
value of the water-flow was nearly half a gramme per second in all cases, and the
results were reduced to a value of Q 0 = 5 calories per second to render the
experiments with the different conductors strictly comparable. The actual heat-loss
from the flow- tube could not be measured by this comparatively rough method, but I
made some attempts to obtain comparative estimates of the temperatures of the
outside of the flow-tube by winding a platinum wire round it, covering the spiral
with flannel, and observing its resistance. These measurements could not lay claim
to any accuracy, but were useful as an indication of effects to be expected.

Table YI. — Superheating of Stranded Conductor.

Form of Conductor employed.

Conditions of Flow.

Superheating.

Five strands, irregular .

. . Mixed

0

2-8

Same, annealed and straightened .

Linear

6-5

' ., twisted into a rope ....

Linear

8-5

,, spiral fitting tube ....

. . Mixed

3-0

In the first case the wire was taken as it came from the reel. The strands were
well separated, and crossed each other irregularly, so that the colour-band was
completely broken up and mixed to a uniform tint in a space of 10 or 15 centims.
The flow was not precisely turbulent or eddying, but the stream-lines were so quickly
sub-divided and mixed that the same effect was produced. When the wire was
annealed and straightened, # the colour-band remained practically unbroken from end
to end of the tube, but as the strands were still separate, the heat was more or less
distributed over the cross-section. The twisting of the wire into a rope in the third
experiment diminished the effective surface and increased the superheating, the
value of which closely approached that calculated for a central conductor in a
2 millim. tube assuming the conductivity of water ‘0016 C.G.S. The great
diminution of the superheating in the fourth case, in which the wire was wound into

* It is probable that the failure of Dr. Barnes to obtain consistent results with a stranded conductor
may have been due to the use of annealed wire, which would inevitably become straightened in fitting up
the apparatus unless special care were exercised. I was not aware of this mistake at the time.

118

PKOFESSOR HUGH L. CALLENDAR OX

a spiral of about 1 centim. pitch, closely fitting the tube, was chiefly due to the
breaking up of the stream-lines. The colour-band rapidly became mixed to a
uniform tint as in the first experiment. I found, however, from the indications of
the platinum thermometer wound round the outside of the tube, that the glass was
considerably heated by the close contact of the spiral. The simpler arrangement of
the loose stranded conductor was equally effective in mixing the stream-lines, and
appeared to be free from this defect.

An excellent illustration of the possible effects of a faulty arrangement of the
conductor is given by Dr. Barnes in § 6, p. 234. In this case it is possible to calculate
the actual heat-loss from a knowledge of the correct value of J for the temperature
of the experiment.* The conditions were purposely chosen to exaggerate the errors
as much as possible, and it must not be imagined that such large differences could be
obtained without special pains in the arrangement of the heating conductor. The
normal heat-loss for this calorimeter at a temperature of 26° with a constant gradient
of temperature along the flow-tube was approximately ‘070 watt per degree rise.
Starting from this value, it is possible to calculate the limiting values of the heat-loss
for either condition of flow, that of the metallic tube or the small concentric con¬
ductor, by drawing the curves representing the actual distribution of temperature in
either case, and making a suitable allowance for the loss of heat from the thermo¬
meter. Dr. Barnes has given a pair of imaginary curves for these two cases in
§ 2, p. 152, but it should be observed that these curves are not drawn to scale,
being merely intended to illustrate the general nature of the difference, which is
considerably exaggerated in order to make it clearer.

Table VII. — Superheating of Straight Conductor in 3 millim. Flow-Tube.

Number of
experiment.

Position of
conductor.

Flow,

Q-

Watts,

EC.

Superheat of
wire.

Heat-loss,

observed.

Limit,

calculated.

(1)

"1 At side of f

•600 .

21-7

o

8-0

•1108

•1140

(2)

J tube t

•277

10-4

4-1

•0909

•0920

(3)

I In middle of f

•600

21-7

12-8

•0482

•0530

(B

J tube \

•271

10-7

7-0

•0773

•0610

The observed values of the superheat and of the heat-loss agree in a general
manner with those calculated by the theory given in § 32, but there are some
difficulties. The limit in the last column of lines (1) and (2) is estimated for a
metallic flow-tube, and it is difficult to see how the heat-loss for a glass flow-tube
could approach this value so closely when the conductor is in imperfect contact with
the glass over a small fraction of its surface. The superheat 12°'8 in line (3) agrees

* In discussing these observations Dr. Barnes takes (EC - 4-2 Q dd) in place of the actual heat-loss,
which unduly exaggerates the difference. The conductor had evidently been annealed , otherwise it would
have filled the tube instead of resting along the side.

CONTINUOUS ELECTRIC CALORIMETRY.

119

very closely with the value 13°'3, calculated for a similar case in § 32, hut the heat-
loss is less than the limit for a small conductor in a large tube. On the other hand,
the observed heat-loss in line (4) is greater than the normal value ’0700 instead of
being less, as theory requires. It is possible that there may have been some other
source of error in this experiment due to shifting the thermometer in the outflow-
tube or similar causes.

When the first apparatus was set up with the vacuum-jacket in June, 1897, only
three strands of '006/' wire were employed, as it was necessary that the resistance of
the heating conductor should be of the same order as that of the manganin current-
standard. The latter had been made of 1 ohm resistance to suit the mercury
experiment, and there was not time to make another, as the experiment had to be
tried before the meeting of the British Association at Toronto. In order partially to
compensate for this defect, I took pains to exaggerate the irregularity of the wire by
bending it into a zigzag before pulling it through the tube, but the superheating of
the wire (4° at 14 watts) proved to be somewhat excessive. In spite of this the
readings were extremely steady, and could be taken easily to 2 or 3 parts in 100,000.
The accuracy of this preliminary series of experiments was seriously impaired by the
bad fitting of the thermometers in the inflow- and outflow-tubes, but the results
showed that the method was capable, under suitable conditions, of attaining a very
high order of precision on account of the great steadiness of the readings, which was
much- more perfect than anyone could have anticipated without actual trial.

Shortly after the British Association meeting, another manganin current-standard
of 1 ohm resistance was made, and placed in parallel with the first. The platinum
conductor, was composed of 6 strands of •006" wire, which gave a much better
distribution of the heat. After numerous preliminary difficulties of temperature
regulation, of leakage, and of bad fitting and connections had been overcome, the
results, though -often very consistent, still showed occasional discrepancies, especially
after refitting. I was inclined to attribute these difficulties at the time either to bad
fitting of the thermometer in the outflow-tube as previously explained, § 26, or to
variable contact between the conductor and the walls of the glass tube, or possibly to
bad solder joins on the manganin current-standards, one of which had actually broken
away on one occasion.

With the view of preventing the uncertainty of contact with the walls of the glass
tube, I thought at one time of trying conductor in the form of a twisted strip,
which I had employed some years previously in experiments on the viscosity of
liquids by the Wheatstone-bridge method, for which I required a continuously
variable resistance analogous to a slide-wire in the electrical method. 1 found that a
round wire sliding in a tube would not do for this purpose owing to the impossibility
of centering. A twisted strip, sliding in a tube which it closely fitted, evaded this
difficulty, but led to further anomalies, which proved, on investigation by the colour-
band method, to be due to the fact that the motion of the liquid became turbulent if

PROFESSOR HUGH I. C ALLEN I>AR OX

120

the pitch of the twist was too steep. This change in the character of the flow was a
serious defect in the viscosity experiment, but was exactly what was required for the
calorimeter. As the calorimeters were already made, and I could not choose the
flow-tube to fit the strip, I procured a special pair of rolls about the end of 1897 for
rolling a wire to fit the flow-tube as closely as possible. But after carefully rolling a
wire to fit one of the calorimeters at one end, 1 found that it would not go through
the tube, on account of want of uniformity of diameter, and in particular on account
of a join near one end which unduly constricted the bore. I accordingly abandoned
the attempt at the time, as T did not feel at all certain of the necessity of making
any change in the form of the conductor. But the twisted strip was subsequently
employed by Dr. Barnes at my suggestion with great success in two other calori¬
meters, the tubes of which were probably more uniform.

The method independently devised by Dr. Barnes for eliminating the effect of the
stream-lines, and at the same time preventing all contact between the conductor and
the glass, was to wind a rubber cord round the conductor so as nearly to fit the tube,
after a similar manner to that employed in fitting the copper sleeve of the outflow
thermometer. This is undoubtedly a very simple and effective method, and unlikely
to get out of order. It is quite possible that some of the discrepancies in the earlier
experiments may have arisen from the accidental disarrangement of the stranded
conductor, which might be pulled tight along the side or middle of the tube in
refitting the apparatus, either of which contingencies would lead to serious errors.
There were many other sources of error and difficulties in the earlier experiments
w hich might have accounted for the effects observed, but Dr. Barnes’ opinion on this
point is entitled to the greatest weight, as lie was personally responsible for the
greater part of the fitting-up of the apparatus. With proper care I have no doubt
that it would be possible to obtain as accurate results with the stranded conductor as
by any other method, but I should be inclined to prefer the rubber-spiral method as
being safer and more certain.

A possible objection to the rubber-spiral is that, since the wire is held central and
considerably superheated, the temperature of the surface of the glass must necessarily
be less than the mean of the flow, in spite of the mixing of the stream-lines. This
would not matter if the difference of temperature were independent of the flow. Since
it is impossible to calculate what the effect would be in the case of turbulent flow, the
question can be answered only by trial. The experiments with the twisted strip,
which was partly in contact with the glass and gave a higher heat-loss, were under¬
taken, at my suggestion, with the object of testing this important point. The
extremely close agreement of the results with those obtained with the rubber -spiral
at 80° C. are sufficient proof that the error, if any, must be extremely small. It is
unlikely that it could amount to more than two or three parts in 10,000 at 30°,
and there is no reason to think that this type of error should increase largely at
higher temperatures, where the viscosity of water is much smaller and the conductivity
probably much larger.

CONTINUOUS ELECTRIC CALORIMETRY

121

(34.) Correction for Variation of the Temperature-Gradient in the Floiv-Tuhe.

The elementary theory of the elimination of the heat-loss in the steady-flow method
of calorimetry assumes that, if the electric current and the flow of liquid be simul¬
taneously varied in such a manner as to keep the rise of temperature the same, the
heat-loss by radiation, &c., will remain constant. Dr. Barnes (p. 225) has quoted
experiments to show that this condition is very closely satisfied in the present method,
and has calculated all the results of the investigation on this assumption. It will be
noticed, however, that there are small systematic divergences in the experimental
verification for the small flows, which, though amounting only to a few parts in 10,000,
require careful examination as possible indications of constant errors.

So long as the distribution of temperature throughout the apparatus is accurately
the same for the same rise of temperature, whatever the flow, the heat-loss must also
be identical. But if there is any systematic change in the temperature distribution
with change of flow, then there must be a corresponding systematic difference in the
heat-loss, which will lead to constant errors in the calculation if no account is taken
of it. A possible source of error of this type is loss of heat by conduction along the
outflow-tube. When the flow is large, the heated liquid passing along the tube will
keep it nearly at a uniform temperature, so that the gradient in the outflow-tube will
be small, and the conduction loss correspondingly minute. As the flow is diminished,
supposing the temperature of the outflow to remain the same, the gradient in the
outflow-tube must increase in proportion to the reciprocal of the flow, since the
radiation-loss remains nearly the same. The conduction-loss will vary directly as the
gradient, ,or inversely as the flow, for a given rise of temperature.

A small error of this kind, due to conduction, was detected at an early stage in the
mercury-calorimeter, owing to the large mass of mercury in the flow-tube, the small
rate of flow, and the relatively high thermal conductivity of the liquid. It was
practically eliminated by filling the greater part of the outflow-tube from the end of
the vacuum-jacket with paraffin wax, leaving only a small passage for the outflow of
mercury. This made the conduction-loss very small, and nearly independent of the
flow. In the water experiment it is easy to see that the conduction loss must be
practically negligible in any case, but special pains were taken to make it as small as
possible, and to verify its non-existence.

A more important correction of this type is that due to variation of temperature
gradient in the fine flow -tube, which can be estimated with considerable precision.
As the liquid flows along the tube it is receiving heat at a nearly uniform rate from
the electric current, but it is also losing heat more and more rapidly by radiation as
its temperature rises. As a result, the mean temperature of the fine flow-tube, upon
which the radiation-loss chiefly depends, usually exceeds the mean between the initial
and final temperature by an amount which varies, to a first approximation, inversely
as the flow.

VOL. CXCIX. - A.

R

122

PROFESSOR HUGH L. CALLENDAR ON

In order to calculate the value of this correction in terms of the heat-loss and the
flow, it is necessary to consider the differential equation of the distribution of tem¬
perature in the flow-tube. We assume, as a first approximation, that the temperature
of the surface of the flow-tube, on which the loss of heat depends, is at each point
nearly the same as that of the liquid flowing through it. This is very nearly true in
the case of mercury, and to a sufficient approximation in the case of water under
suitable conditions.

The rate of evolution of heat by a current C amperes in a length dx of the tube is
C2rdx watts, where r is the resistance in ohms per centim. The rate of loss of heat
is fdp dx, where p is the perimeter of the flow-tube in centims., and f the emissivity
in watts per sq. centim. per degree of temperature excess d. The rate of gain of heat
by the liquid is JsQ dd, where J is the number of joules in one calorie ; s the specific
1 1 eat of the liquid in calories per gramme degree C ; Q the flow of liquid in grammes
per second ; and dd the rise of temperature in a length dx. Since it is necessary to
take account of the change of resistance with temperature, we must substitute for r
the value rQ (1 + ad), where a is the temperature -coefficient of the increase of
resistance, and r0 the value of r when 0=0. We thus obtain the linear differential
equation

J.sQ dd/dx -f- ( fp — C 2cir0) d = CV0 . (1).

Writing for brevity, A = ( fp — C2«r0) / JhsQ, and B = CV0/JsQ, and observing
that d — 0 when x — 0, since the liquid flows into the tube at the jacket-temperature,
the solution of this equation is

d = (l - <rAx) B/A . (2).

Since A is very small compared with B in the actual experiment, we may obtain a
sufficient approximation for our purpose by expanding the exponential and neglecting
the terms beyond A2, which gives

dx — B.r (1 — Ax/2)

If the whole length of the fine flow-tube is l, the temperature at the end of the

flow-tube will be approximately

0i = Bl (1 - M/2) . (4),

and the mean temperature dm from 0 to /, will be

d m = B/ (1 - AZ/3)/2 . (5).

It will be observed that the value of A is zero, and that the gradient is constant
throughout the tube for a particular value of the current, C2 = fp/cir0, which may be
called the critical value of the current. In this case the radiation-loss along the

CONTINUOUS ELECTRIC CALORIMETRY.

123

flow-tube is exactly compensated by the increase of resistance of the conductor with
rise of temperature. I made use of this relation in some experiments on conduction
of heat in metals by an electrical method with Mr. King in 1895, and also in some
experiments on the conduction of heat in liquids, § 31, in which the elimination of the
heat-loss was a matter of some importance as simplifying the differential equation.
In the calorimetric experiment, the constancy of the gradient along the tube was
not a matter of primary importance, provided that the temperature distribution was
approximately the same for different flows. Moreover, the relation could not apply
accurately to both the flows required in the experiment. Nevertheless, I thought it
worth while, in designing- the dimensions of the calorimeter and the details of the
experiment, to arrange that the compensation might hold for a value of the flow
between ‘5 and l'O gramme per second in the water experiment, as nearly as it
could be estimated beforehand. The gradient would then be nearly constant, and
the mean temperature of the flow-tube nearly half the rise of temperature observed
with the differential-thermometers.

(35.) Application to the Mercury Experiments.

Neglecting for the present that part of the heat-loss which occurs in the outflow-
tube before the liquid reaches the middle of the thermometer bulb where its
temperature is measured (which loss is a comparatively small fraction of the whole
in the mercury experiment), the systematic error of the elementary theory given by
Dr. Barnes, p. 152, consists in assuming that the mean temperature of the flow-tube
is always the same for the same rise, or that the gradient is indejiendent of the
How. This is equivalent to assuming the mean temperature of the flow-tube equal
to 6i/ 2 instead of dm. The error of this assumption may be approximately estimated
from equations (4) and (5) above, which give

6», ~ (1 + A.l/6) 0//2 . (6).

If we write (as in Barnes, p. 242) dd for the whole rise of temperature observed
by the differential thermometers, and h dO for the heat-loss, we must then regard h
as variable with the flow, since the heat-loss is really fpid,„. We thus arrive at the
expression,

Heat-loss = fpldm = fpl (1 fl- A//6) di/ 2 = h0 (1 fl- A//6) dd . . . (7),

in which dd is written for d(, and h0 is the value of the heat-loss per degree rise
when the gradient is constant (A = 0), namely fpl /2. 1 We have also, to the same
order of approximation,

A l = 2/<0/J.5>Q — a dd
R 2

(8).

124

PROFESSOR HUGH L. CALLEXDAR OX

Hence, the complete equation of the method, when corrected for the effect of the
variation of the temperature gradient in the fine flow-tube, becomes

EC = J.s-Q <19 + h0 (1 + hjBJsQ - a cld/6) <16 . (9).

If we divide through the equation by cI6, and write s = s0 ( 1 -f- cl), since our
object is to determine the variations of s, we obtain

EC jcl6 — Js0Q = = J.s0Q d -j- /?0 (l — a c 16/6) + h02/BJsQ . . (10),

in which D is employed as an abbreviation for the expression on the left-hand side,
which it is most convenient to calculate as the first stage in the reduction of the
observations.

By combining the observations for two different flows, Q' and Q", for which the
current is adjusted to give approximately the same rise of temperature, cW, if D' and
D ' are the corresponding values of the difference observed, we obtain

(IT - D")/(Q' - Q") = J s,d - V/3J,Q' Q" . (11).

Since the last term, which represents the effect of the correction sought on the
variations of the specific heat, is generally very small, the equation may be readily
solved by approximation, employing the value of h found in the first instance bv
neglecting the term involving Jr.

As an example of the order of magnitude of the correction, and of the method of
application, we may take the following experimental results, which were given as an
illustration of the method at the meeting of the British Association in 1897, and
were quoted in the ‘ Electrician ’ of that date. In all the earlier series of observa¬
tions, three independent flows were taken, with the same rise of temperature at each
point, with the object of verifying the theory of the method, and detecting possible
sources of error. If the values calculated from the largest and smallest flow
disagreed with the observation on the intermediate flow, it was a sure sign of some
error or defect in the work. I thought at first that it might be possible to determine
the conduction error experimentally in this manner, but the effect proved to be
too small.

Table VIII. — Example of Calculation of Specific Heat of Mercury. #

Flow, Q.

Rise, <16.

Watts, EC.

EC/cW.

• 1400Q.

I).

A2/3J*Q.

Results,

corrected.

(1)

8-753

11-764

14-862

1-2632

1 • 2255

•0377

•0008

/i= -0546

(2)

6-740

11-8720

11-696

•9851

•9435

•0416

•0010

Jsd= - -00202

(3)

4-594

12-301

8 • 488

•6901

•6433

•0468

•0015

J.s= -13798

* These results are given merely as an illustration of the method : they were not corrected for the
absolute values of the resistances and other minor sources of error.

CONTINUOUS ELECTRIC CALORIMETRY.

125

In working out these results, the standard value of the specific heat was taken
as Js0 = ’1400 joule per gramme degree, since the specific heat of mercury is
approximately -g^th of that of water. If we take only the values of the difference D
corresponding to the first and third flows, and calculate the values of h and J sd,
according to the elementary theory, neglecting the correction term Id/dJsQ, we find

J sd = (D' - D")/(Q' - Q") = - -0091/4-159 = - ‘00219,
whence, h = "0568, J s = -1400 — "00219 = "13781 joule per gramme degree.

As a verification, we may calculate the value of D for the intermediate flow (2).
We find Ih = "0420, in place of the observed value "0416. The difference is only
four parts in 10,000, on EC/c/d, and might well be attributed to errors of obser¬
vation in these preliminary experiments.

Inserting the correction for the variation of the temperature gradient in the flow-
tuhe, by subtracting li2J 3JsQ from each of the corresponding values of D, and then
calculating as before, we find the corrected results given in the last column, which
exceed those calculated on the elementary theory by nearly one part in 700. It
should be remarked that this correction can be deduced with certainty from (1)
and (3) without reference to (2). It cannot be calculated satisfactorily from three
flows, as it depends on small differences.

(36.) Correction of Results with Water Calorimeter.

In applying the correction to the water calorimeter, it is necessary to take some
account of the heat-loss from the outflow- tube round the thermometer bulb, as well
as that from the fine flow-tube. This changes the numerical factors, which depend
to some extent on the dimensions of the tubes, but the theory of the correction
is otherwise unchanged. Assuming the heat-loss from the thermometer bulb to
be two-fifths of the whole, which is sufficiently exact, since the whole correction is
very small and a change of one-tenth in the ratio would not alter the result by more
than 2 or 3 per cent, of itself, equation (10) becomes

EC /tie - Js0Q = D = Js0Qd + h0( 1 - add/ 10) + lif 1 l/25JsQ . . (12),

and the correction to be added on this account to the value of the specific heat in
joules, as calculated by Dr. Barnes, is given by the expression

Correction for Variation of Gradient with Flow = -f- 1 l/)02/25J*-Q,Q,/ . (13)

In calculating this correction it is desirable to use the (corrected value of h 0, which
is obtained from that of h as given by Dr. Barnes by adding the terms

Correction to value of h = + ahdO/10 — (11 A~/2 5 -J.s) (1/Q' -f 1/Q") . (14).

126

PROFESSOR HUGH L. C ALLEND AR ON

The standard value of Js0 assumed in reducing the tables in the case of water for
calculating the value of D, was the British Association unit of 4‘200 joules. Inserting
this value, and jmtting a — '0039, we have the numerical formulae

Correction to h = + ‘00039 hdO — T05/i2/Q' — T05/i3/Qr/.

Correction to J = + T05/i03/Q'Q" . (15).

These formulae are to be employed in correcting the results for the specific heat of
water given in the summarized tables (Barnes, p. 243).

In cases where more than two flows are available at the same time and under the
same conditions, it is naturally possible to obtain a more reliable value for the result
by utilising all the Hows, so as to minimize the effect of accidental errors. The
accuracy of the work may then be verified by comparing the observed values of D for
each flow with those calculated by equation (12). As an illustration we may take
observations I. and II. with calorimeter D (Barnes, p. 243), which are also selected
by Dr. Barnes as an experimental verification of the elementary theory.

Table IX.— Correction for Variation of Gradient in Flow-Tube.

Specific Heat of Water. Observations I. and II. (Barnes, Table XVIII., p. 243),

Calorimeter, D.

Number of flow

(1)

C)

(3)

(4)

(5)

Mean of 2, 3, 4

Flow, Q, gramme per second

•6741

•3993

•3902

•4967

•2482

•4287

ho, watts per degree

•06975

•06975

■06975

■06975

•06975

•06975

■0039h0cW/l0 ....

- -00021

- -00022

- -00022

- -00022

- -00022

- -00022

• 105/n2/Q .

+ -00076

+ -00128

+ -00132

+ -00103

+ -00207

+ -00120

.

Oi

O

r-H

o

1

- -01281

- -00758

- -00741

- -00944

- -00471

- -00815

D, calculated ....

(-05749)*

•06323

•06344

•06112

•06689

(•06258)*

D, observed . .

•05749

•06317

•06318

•06135

■06691

•06257

■ 07 1 45-- 00207 Q . . .

(-05749)*

•06318

•06337

•06117

•06631

(•06257)*

The last column gives the mean of the three intermediate flows, which has been
combined with the first flow in the calculation of the results. The second line gives
the approximate values of the flows, which are required in working out the results.

The next four lines give the values of the several terms in equation (12) for the
calculation of D. The two unknown quantities h0 and J sd are calculated by assuming
the values of D enclosed in brackets for (l) and the mean of (2, 3, and 4).

The values of D “ calculated ” are obtained by adding the numbers in the four
lines above, and are corrected for the effect of variation of the gradient in the fine
flow-tube. The values D “observed,” given in the next line, are taken from the
tables, and are found by subtracting 42Q from EC/dO.

* Assumed in the calculation of h0 and d.

CONTINUOUS ELECTRIC CALORIMETRY.

127

The last line contains the values of D calculated on the elementary theory,
neglecting the variation of the gradient, assuming the same two values (1) and
(2, 3, 4), and taking the heat-loss per degree constant. This assumption gives
h — ‘07145 for the heat-loss, and J scl = — ‘00207 joule per gramme degree for the
defect of the specific heat from 4‘200.

Comparing the observed and calculated values of D on the two assumptions, we
see that the relatively large discrepancy of ‘00060 on the elementary theory in the
value calculated for the small flow (5), is exactly accounted for by the correction for
variation of the gradient in the flow-tube.

The effect of applying the correction in this particular case is to increase the value
of the specific heat from 4 1793 to 4 ' 1 8 1 0 , i.e., by 4 parts in 10,000. This correction
is very small, but it cannot be neglected, because it is systematic. Besides, it is
much larger than the errors of observation, which rarely amount to so much as
1 in 10,000 on a single flow with a rise of 8°.

In the discussion of these observations, Dr. Barnes (§ 5, p. 226) concludes, from
the close agreement of the observed and calculated values of the heat-loss for the
larger flows, that the elementary theory is valid and requires no correction, provided
that the flow exceeds a certain limit, about ‘35 gm./sec. But the agreement for
the larger flows results merely from his method of calculation, and is no evidence
that the correction is negligible. The need for the correction could not arise per
salturn below a certain limit, as he suggests.

The existence of the correction is obviously required by theory, and it is a
remarkable verification of the accuracy of his observations that the small apparent dis¬
crepancy on the smaller flows, which he was prepared to attribute to some unknown
source of error, should be so nearly accounted for by the variation of temperature
distribution with flow, which is a necessary consequence of the method adopted.

In going through the summary of observations (Barnes, Table XVIII.), it may be
noticed that there is generally a small systematic error of this nature in the results,
as calculated on the elementary theory, tending to make the value of the specific
heat smaller, the smaller the flows from which it is calculated. It would not,
however, be worth while to recalculate the whole in detail as above illustrated,
because the correction is so small that it may reasonably be applied to the observa¬
tions as a whole.

In any particular case, for a pair of flows, the correction can most easily be applied
by means of the numerical formulae (15) already given. This will not give the best
results, if more than two flows are available, but the residual errors will then be
accidental, so far as this particular correction is concerned. A few examples of this
method of correction, taken from the first few experiments at 29°‘10 C. with
calorimeter C, are given in the following table. The heat-loss in this calorimeter
was much smaller than in calorimeter D, owing to a better vacuum and a smaller
flow-tube. The correction is consequently smaller, and its effect is more obscured by

128

PROFESSOR HUGH L. CALLER DAR ON

accidental errors, but so far as it goes, it tends to bring the observations into bettei
agreement.

O

Table X. — Correction for Variation of Gradient in Flow-Tube, Calorimeter C.

Number of
Experiment.

Values of Flow.

Heat-Loss li per 1°.

Specific Heat in Joules.

Q'.

QA

Table.

Reduced.

Table.

Corrected.

III.

•665

•399

•04914

•04S59

4-1803

4-1813

IV.

•501

•258

•05123

•04989

4-1771

4-1791

V.

•660

•392

•04937

•04859

4-1795

4-1805

VI.

•590

■ 375

•04965

•04882

4-1790

4-1802

The general effect of the correction here is to raise the values by about 1 in 4000
for the larger pairs of flows. The discrepancy between Experiments III. and IV. is
mainly chie, as the values of D show, to accidental errors of the small flows in
opposite directions. The mean value 4H803 agrees very well, allowing for the
difference of temperature, with the value 4-1810 deduced from observations I. and II.
with calorimeter D at 28°'0 C., which would give 4'1805 at 29 'TO C.

(37.) Variation of the Gradient-Correction with Temperature.

The importance of this correction arises chiefly from the fact that the value of the
heat -loss increases with rise of temperature for any calorimeter. As a result , the
correction, which depends on the square of the heat-loss, is considerably larger at
higher temperatures, and thus affects the curve of variation of the specific heat as
well as the absolute values. If the correction were constant, it would affect only the
absolute values, which would matter little owing to the uncertainty of the electrical
units.

As an illustration of the greater importance of this correction at higher
temperatures, the last four observations (Barnes, Series 8, Table XVIII.) are given in
the next table, reduced and corrected in a similar manner. Although these observa¬
tions do not include any exceptionally small values of the flow, the correction at the
higher points nearly reaches 1 in 1000 owing to the larger value of the heat-loss.

CONTINUOUS ELECTRIC CALORIMETRY.

129

Table XI. — Correction for Variation of Gradient in Flow-Tube, Calorimeter C,

Series 8.

Number of
Experiment.

Temperature,

Centigrade.

Values of Flow.

Heat-Loss per 1°.

Specific Heat in Joules.

Q'.

Q".

Table.

Reduced.

Table.

Corrected.

LII.

o

74-05

•621

•384

•08469

•08198

4-1920

4-1950

LIII.

91-55

•644

•402

•10011

•09643

4-2017

4-2055

LIY.

80-38

•617

•388

•09218

•08891

4-1951

4-1986

LV.

68-21

•603

•388

•08283

•08011

4-1890

4-1919

The effect of the correction on the curve of variation of specific heat is most
readily appreciated by taking the approximate formula for the variation of the heat-
loss with temperature, given by Dr. Barnes, p. 253, which, when slightly reduced to
allow for the correction of the value of h given in equation (15), becomes

h = -0300 + -00070/ . (16).

This formula represents the experimental results very fairly, except for the lower
temperatures in the neighbourhood of 0° C. It leads to the following values of the
corrections for the specific heat in joules and calories.

Table XII. — Variation of Gradient-correction with Temperature, and Corrected

Values of Specific Heat of Water.

1

Temperature,

Centigrade.

°

Heat-Loss.

Correction T05 ho2/ Q'Q".

Results,

1900,

Corrected.

Reduced
to H scale.

Formula (16).

E0*

Joules.

Calories.

o

0

•0300

•033

•00038

•00009

1-0080

1-0084

10

•0370

•039

•00058

•00014

1-0029

1-0031

20

•0440

•045

•00082

•00019

1-0000

1-0000

30

•0510

•051

•00109

•00026

0-9987

0-9986

40

•0580

•058

•00141

•00034

0-9986

0-9984

50

■0650

•066

•00177

•00042

0-9993

0-9989

60

•0720

•074

•00218

■00052

1-0005

1-0000

70

•0790

•083

•00263

•00063

1-0018

1-0013

80

■0860

•094

•00312

•00074

1-0033

1-0027

90

•0930

•105

•00364

•00087

1-0048

1-0041

100

•1000

•117

•00420

•00100

1-0062

1-0055

In calculating the correction, the value of the product Q'Q,y is taken as
'64 X ‘39 = -25, which represents fairly the values of the larger pair of flows on
VOL. CXCIX. — A.

s

130

PROFESSOR HUGH L. CALLENDAR ON

which the results mainly depend. The corrections calculated from the actual
numbers in the tables are, as a rule, slightly larger than those calculated from the
formula for the heat-loss, but the latter are usually within 1 in 10,000 of those
observed.

It may be interesting to remark that formula (16) gives a mean rate of increase of
the heat -loss which is proportional to the fourth power of the absolute temperature
over the range 0° to 80°. The fourth power law is well known to represent heat-loss
by radiation with considerable accuracy over a moderate range at ordinary
temperatures under conditions similar to those of this experiment. Values of the
1 eat-loss calculated according to the fourth power law starting from the same value
at 30° C., are given, for comparison, in the third column, headed E0h They
represent the experimental numbers rather better than the linear formula (16) at
temperatures below 30°, but give results which are a little too high at 80° and 90°.
The corrections in the table have been calculated from the linear formula, but the
difference would be very slight over the greater part of the range.

The values of the specific heat given in the column headed “Results, 1900,
Corrected,” are obtained from those calculated by the formuke given by Barnes,

‘ Proc. Roy. Soc.,’ 1900, p. 242, by adding the correction in calories given in the
previous column, and expressing the results in terms of a unit at 20° C., instead of
the unit at 16° employed by Barnes. Although the values of the corrections are
worked to the next figure, I have not considered it desirable to give the values of
the specific heat beyond 1 jiart in 10,000.

It happens, by a curious coincidence, that the correction for the variation of the
temperature gradient is very nearly equal and opposite to the correction required to
reduce the results to the hydrogen scale, if calculated, as previously explained in
Section 23, from the observations of Joule and Thomson for air at a constant
pressure of 76 centims. The values given in the last column, reduced to the
hydrogen scale, are practically identical with those calculated directly from Barnes’
formulae without correction, except that they are expressed in terms of a unit at
20° C., and are corrected for an obvious misfit of the formulae at 55° C. (see §46).

Part V. — Discussion of Results.

(38.) Meaning of the Term “ Specific Heat.”

The term “ specific heat ” is here employed as an abbreviation for the phrase
“ specific capacity for heat,” or “ thermal capacity of unit mass, ’ i.e., the quantity of
heat per unit mass per degree required to raise the temperature of a substance.
In a similar manner, “specific electrical resistance” or “ resistivity” of a substance is
understood to mean the resistance of the material per unit area of section per unit
length. On this understanding, specific heat may be measured in terms of any con-

CONTINUOUS ELECTRIC CALORIMETRY.

131

venient unit, either in joules per gramme per degree Centigrade, or in foot-pounds
per pound per degree Fahrenheit, without any implied reference to the properties of
water. On the other hand, it is not unusual in elementary text-books to define specific
heat as the ratio of the thermal capacity of a given mass of the substance considered
to that of an equal mass of water at a standard temperature. The two definitions
lead to the same numerical results, provided that the unit of heat is the thermal
capacity of unit mass of water at the standard temperature. But there is nothing in
the derivation of the term “ specific heat ” to imply that it denotes a ratio with
respect to water at a standard temperature, and I think that this definition unduly
restricts the meaning, and has given rise indirectly to a good deal of confusion.

(39.) Choice of a Standard Temperature for the Thermal Unit.

In addition to the absolute unit of heat, which is naturally the same as the unit of
mechanical energy, it is necessary for practical calorimetry to adopt as a standard of
reference a thermal unit equal to the quantity of heat required to raise unit mass
of water one degree at a standard temperature. For purely academic purposes, it
would suffice to adopt either of the time-honoured standards at 0 C. or 4° C., which
have been frequently proposed, and are still to be found in the majority of text¬
books. But it has been conclusively shown that the specific heat of water at these
low temperatures is considerably higher than over the range which is commonly
employed for calorimetric determinations. The units at 0 and 4° would be practically
inconvenient on this account. A still more serious objection is that the specific heat
at these temperatures cannot readily be determined with the same order of accuracy
as at ordinary temperatures. The unit of 4-200 joules proposed by the British
Association Committee, which was supposed at that time to represent the specific
heat of water in joules per gramme degree at 10° C., is open to the additional objection
that it is really an absolute unit in disguise, and that, as such, it is superfluous, and
does not satisfy the requirement of a thermal unit defined in terms of the specific
heat of water at a definite temperature.

For practical purposes, it is evidently necessary to define the thermal unit in terms
of the mean specific heat over a range of temperature rather than at a definite point.
The range of 1°, which is generally taken in definitions, is evidently too small for
the accurate measurement of the rise of temperature in terms of the fundamental
interval. These very small ranges of temperature have frequently been employed in
calorimetry, as in Joule’s earlier experiments, for the purpose of reducing
the uncertain correction for external heat-loss ; but with modern appliances for
accurate thermal regulation, and provided that the duration of the experiment is not
unduly prolonged, it is quite possible to employ a rise of 10° or more without the
uncertainty of the heat-loss exceeding the probable theormometric error.

So far as the thermometric error alone is concerned, the obvious interval to select

S 2

132

PROFESSOR HUGH L. CALLENDAR ON

is the fundamental interval itself. For this reason, the “ mean calorie,” which is
equal to one-hundredth part of the quantity of heat required to raise the temperature
of a gramme of water from 0° to 100° C., has been often proposed as the most suitable
thermal unit. But in this case the calorimetric difficulties due to the large range of
temperature are so exaggerated that the relation of the mean calorie to the practical
unit employed in calorimetry cannot he determined with the same order of accuracy
as the practical unit itself can be realized. This point is further illustrated in
Section 45 below. An equivalent proposition is to select as the standard temperature
for the definition of the thermal unit that temperature at which the specific heat is
equal to its mean value over the fundamental interval. The objection to this is that
it leaves the standard temperature uncertain. If, however, a definite temperature of
either 15° or 20° C. were selected, we should have a definite unit, which would
probably be within one part in a thousand of the mean calorie, or near enough for all
practical purposes.

On the whole, a range of 10° appears to be the most appropriate to adopt for the
practical definition of the thermal unit. Howland’s results for the specific heat were
calculated in this manner by taking the mean value over each interval of ten degrees.
Griffiths also adopted a range of ten degrees, 15'’ to 25°, in his experiments, and his
results may be held to refer to the mean of this range, which is also very close to the
mean of the range, 18c to 20°, of Schuster and Gannon’s experiments. For this
reason, I have been in the habit for some years of expressing results in terms of the
mean specific heat of water over the range 15° to 25° C., which is, within 1 part in
20,000, the same as the value at 20° C. The mean thermal unit over this range may
conveniently be called the “calorie at 20° C.,” although of course it cannot be
practically realized except as the mean over a range.

The preliminary results for the variation of the specific heat of water communicated
to the British Association at the Dover meeting in 1899, were expressed in terms of
the calorie at 20° C. I have thought it important to retain this unit in the present
paper to avoid confusion, although Dr. Barnes and Mr. Griffiths* have since proposed
units at 16° and 15° respectively, as being probably closer in magnitude to the mean
calorie between 0° and 100° C. From a scientific point of view there is little to
choose between these units, and the relation between them is known with a hiadi
degree of accuracy ; but as a question of practical calorimetry, I think the unit at
20° is undoubtedly superior. In all accurate calorimetric work it is necessary to
employ thermal regulators, and there can be no doubt that from this point of view
the range 10° to 20°, corresponding to the unit at 15° C., is too low. The range 20°
to 30° would be better for temperature regulation than 15° to 25°, but I adopted the
latter as corresponding to the mean of Howland’s, and Griffiths’, and Schuster’s
experiments, and as agreeing more nearly with the mean range of an ordinary

* More recently, Griffiths (‘Thermal Measurement of Energy,’ Cambridge, 1901) has proposed to
adopt the calorie at 17 ‘5° C., or the mean value over the range 15°-20° C.

CONTINUOUS ELECTRIC CALORIMETRY.

133

calorimetric experiment without a thermal regulator. It might be objected that
Schuster (range 18° to 20c) and Ludin (range 11° to 18°) found it necessary to keep
the calorimeter always below the temperature of its surroundings, in order to avoid
readings on a stationary or falling temperature, which are quite unreliable with
a mercury-thermometer, owing to capillary friction. But even in this case it would
be better to use a regulator at 25°, to avoid any risk of cooling the calorimeter below
the dew-point, if a range of 1 0° is required.

(40.) Choice of a Standard Sccde of Temperature.

An equally important question in the definition of the practical thermal unit is the
choice of the scale of temperature to which it should be referred. There would
probably be little hesitation in selecting the scale of the constant-volume hydrogen-
thermometer at 1 metre initial pressure ; but this necessitates the further considera¬
tion of the secondary standard by which the practical unit is to be realized, as it
would of course be quite impossible to employ the hydrogen-thermometer at constant-
volume directly in a calorimetric experiment. The mercury-thermometer, which is
regarded at present as the representative of the normal scale, is a most unsatisfactory
instrument for accurate calorimetric work. Some of its defects in this respect have
already been incidentally mentioned, and it is impossible to regard with confidence
results obtained by its use under conditions so different from those of the comparison
with the ultimate standard. The only satisfactory method of standardizing a mercury-
thermometer under the conditions of experiment, is by comparison with a platinum-
thermometer. In the great majority of cases it would be far less trouble to discard
the mercury-thermometer in accurate work and employ the platinum-thermometer
itself. The results could be nominally reduced to the hydrogen scale in the manner
described in the present paper, with an accuracy which is limited only by the gas-
thermometer. But though nominally expressed in terms of the hydrogen scale, and
subject to all the uncertainties of gas-thermometry, the results would have the
advantage of being really referred to a practical scale which could be reproduced at
any time with an accuracy depending only on that of the original observations.

Although it is very generally admitted that the platinum-thermometer is the most
accurate instrument for scientific work, it is also commonly assumed that the mercury-
thermometer is much easier to work with. This is quite a mistake if an accuracy of
the order of '00 lc C. is aimed at. If anyone wishes to realize the incalculable simplifica¬
tion introduced into accurate work by the employment of platinum-thermometers
instead of mercury, even over the range 0° to 100°C., he should try to perform
an experiment like the present. After endless labour spent in calibrating and
standardizing a suitable series of limited scale-thermometers, he would probably find
all his observations spoilt by uncertainties of stem exposure and capillary friction,
and more particularly by unknown changes of zero at the higher points of the range.

134

PROFESSOR HUGH L. CALLENDAR ON

It is no exaggeration to say that this investigation could not have been carried out
successfully without the direct employment of platinum-thermometers.

(41.) The Work of Regnault.

The work of Regnault (‘ Memoires de lTnstitut,’ Paris, 1847) on the specific heat
of water by the method of mixture at temperatures between 110° C. and 190° C.,
represents the only evidence at present available on the variation of the specific heat
at high temperatures. It was carried out on a large scale with his usual
experimental skill, and is undoubtedly entitled to great weight ; but there are
large discrepancies in several cases between the recorded data and the calculated
results, and very little is known of the scale of the thermometers employed in
the work.

The total capacity of the calorimeter employed was about 110 kilogs. At each
experiment 10 kilogs. of cold water was drawn off, and the same amount of hot
water, at a known temperature, was introduced from a high-pressure boiler. The
rise of temperature varied from 8° to 15°, according to the initial temperature of the
hot water, and was read to the hundredth of a degree. The discrepancies in the
observations taken at any one point under similar conditions are of the order of
•5 per cent. The results have been recalculated from the data columns by
J. M. Gray (‘Proc. Inst. Mech. Eng.,’ 1889), who finds in several cases disagree¬
ments amounting to from 2 to 4 per cent. In all these cases it appears that the
total recorded quantity of water considerably exceeded the capacity of the calori¬
meter. As the result of a careful enquiry, Gray concludes that Regnault’s
calculations are probably right, and that the apparent discrepancies in the data
arise from deficient information or erroneous entries.

It would appear to be hopeless at the present time to make any corrections to the
readings of the thermometers employed for observing the temperature of the hot
water, beyond those which Regnault himself applied. The principal source of
uncertainty lies with the calorimetric-thermometers, as Regnault was unable to
obtain any consistent evidence of deviation from the scale of the air-thermometer
between 0° and 100° C., and did not apply any correction from the mercurial to the
absolute scale. Moreover, no allowance was made for the temporary depression of
zero to which French “ cristal " thermometers appear to be particulary liable. Some
thermometers of this class show a depression of as much as half a degree after
heating to 100° C. If such a thermometer is suddenly immersed in steam, its
reading rapidly rises to a maximum, and then slowly falls for half an hour or so
towards its steady reading. In using a mercury- thermometer in a calorimeter,
where it is exposed to a sudden rise of temperature, the effect of this phenomenon
is to make the observed rise of temperature too large. The maximum reading is
always taken, with additive corrections to represent the subsequently observed rate

CONTINUOUS ELECTRIC CALORIMETRY.

135

of cooling. At the time of reading the maximum, the zero depression has not had
time to produce its full effect, owing to the suddenness of the rise, and it therefore
tends to increase slightly the subsequent rate of fall. In graduating the thermo¬
meter, or comparing it with an air-thermometer, the readings are taken at steady
temperatures, so that the zero depression has time to produce its full effect. These
steady readings will be systematically lower than the instantaneous readings obtained
on suddenly heating the thermometer. Unless the method of variable zero is
employed in taking the observations, it is quite impossible to apply an accurate
correction, since the depression in any given case depends so much on the past
treatment of the thermometer, especially in the case of French “ cristal ” glass. It
is possible, however, to assert that the probable effect would be to produce an error
in the observed rise of temperature approximately proportional to the rise, and
therefore nearly proportional to the excess of temperature of the hot water, since
the same weight of water was employed in all the experiments. The corresponding
error in the value of the mean specific heat deduced would therefore be nearly
constant on the average, although no doubt the variations of the zero depressions in
consecutive experiments may be responsible for some of the individual discrepancies.
The possible limit of error from this source would be about 5 parts in 1000, allowing
for the effect of the zero depression in accelerating the apparent rate of cooling,
which would tend to increase the error. The probable effect would be to make the
specific heat, as calculated by Regnault, about 2 or 3 parts in 1000 too large.

The correction of Regn ault’s thermometers to the hydrogen scale cannot be
applied with any certainty without recovering the original instruments, as different
thermometers of the same glass often differ considerably, and so much depends on
the exact method of treatment. But if we assume Guillaume’s tables for modern
thermometers of similar glass, the correction to the values of the specific heat
would be of -the order of 3 parts in 1000 in the direction of reducing Regn ault’s
results, and would be nearly constant for the different observations.

Including both sources of error, we should infer that Regnault’s values for the
mean specific heat may require to be reduced by a constant correction of 5 or 6
parts in 1000.

Although it is evident that some correction is necessary, I should hesitate to
assume the above estimate without experimental corroboration. 1 find, however, as
explained in the ‘Brit. Assoc. Rep.,’ 1899, that a correction of precisely this order
of magnitude is required to make Regnault’s observations of the mean specific heat
between 20° and 110° C., agree with those of Reynolds and Moorby, between 0°
and 100°, and with those of Barnes, between 40° and 90° C. The correction also
makes Regnault’s observations at 110° agree much better with his own observations
at higher temperatures. It has the further advantage of being the simplest, as well
as the most probable kind of correction to apply. I therefore proposed in 1899 to
adopt Regnault’s formula provisionally for the higher temperatures, merely

136

PROFESSOR HUGH L. C ALLEND AR ON

subtracting the constant quantity '0056 from his values of the specific heat, in
order to make them agree with the curve of variation deduced from the present
investigation at a temperature of 60° C. Thus modified, the formula for the specific
heat s at a temperature t is as follows : —

From 60° to 200° C. s = '9944 + '00004* + *000, 000, 9«2 . . . (l).

Regnault’s formula for the mean specific heat would require further modification,
as it is of course quite erroneous between 0° and 60°. But it is really of compara¬
tively little use to tabulate the mean specific heat. The quantity most often
required is the total heat h from 0° to t. If we adopt as unit the specific heat of
water at 20° C., the total heat from 0° to 60° is 60'020. The value of the total heat
h above 60° is then represented by the formula,

(Above 60°) h = + '220 + '9944* + -000,02$® + ‘000, 000, 3$3 . . (2),

which differs from Regn ault’s formula only in the first two terms, and is deduced
from the formula (l) for the specific heat at t by integration, and addition of a
suitable constant to make the value right at 60° C.

To find the mean specific heat between any two arbitrarily selected temperatures,
which is often required in reducing calorimetric observations, the simplest method of
procedure is usually to take the difference between the values of h corresponding to the
integral values nearest to the extremes of the range, and divide by the whole number
of degrees between the values taken. This will generally give a result which is
accurate to 1 in 10,000. If the range is less than 10° the order of accuracy will
be proportionately less ; but this is immaterial, as the same will probably be true of
the observations themselves with which the comparison is required.

The work of Pfaundler and Platter, of Hirn, of Jamin and Am a fry, and of
many other experimentalists who succeeded Regnault, appeared to indicate much
larger rates of increase than he had found ; but there can be little doubt that the
discrepancies of their results, which often exceeded 5 per cent., were due to lack of
appreciation of the difficulties of the problem. Before the time of Rowland’s
experiments in 1879, sufficient attention had not been paid to the thermometry, and
the results were of comparatively little value.

(42.) The Work of Rowland.

It is unnecessary to give any description or criticism of Rowland’s work, which is
generally recognised as being the most accurate determination of the mechanical
equivalent of the thermal unit at ordinary temperatures. Rowland himself con¬
sidered that his results were probably correct to at least 1 in 500, and that the
greatest uncertainty lay in the comparison of the scale of his mercury-thermometers

CONTINUOUS ELECTRIC CALORIMETRY.

137

with the air- thermometer, which was the most difficult part of the work. His ther¬
mometers have recently been compared with a mercury-thermometer standardized at
Paris, and with a platinum-thermometer standardized by Griffiths, The result has
been to reduce the rate of change of specific heat shown by his original calculations by
nearly one-half, but the absolute value of the specific heat about the middle of the
range remains almost unchanged. The accuracy of his results in terms of the
hydrogen scale has probably been raised to 1 in 2000 on the average by this
correction ; but it must be remembered that the thermometers were compared at
steady temperatures, and not under the actual conditions of the experiment on a
steadily rising temperature. This may affect the validity of some of the conclusions.

In Rowland’s experiments, the most striking difference from ours is the sharp
minimum at 30° followed by a rapid rise. Rowland himself considered that at 30°,
owing to the increasing magnitude and uncertainty of the radiation correction, “ there
might be a small error in the direction of making the equivalent too great, and the
specific heat might go on decreasing to even 40°.” It should be remembered that his
method gives directly the total heat reckoned from the start of each experiment, and
that the values of the specific heat over shorter ranges would be more affected by
thermometric errors, especially near the ends of the range. If our results are com¬
pared with his by means of the total heat starting from the same value at 5° C., it
will be seen that, in spite of the apparent dissimilarity in the forms of our curves for
the specific heat, our values of the total heat do not differ from those of Rowland by
more than 1 in 5000 over the whole range of his experiments. This is shown in the
following table : —

s

Table XIII. — Values of Total Heat of Water compared with Rowland.

Temperature,

Centigrade.

Total Heat,

Callendar and Barnes.

Total Heat,
Rowland.

o

5

5-037

5-037

10

10-056

10-058

15

15-065

15-068

20

20-068

20-071

25

25-065

25-067

30

30-060

30 • 057

35

35-052

35-053

(43.) The Method of Mixture. Ludin.

The experiments of Bartoli and Stracciati (‘Boll. Mens, dell’ Acc. Gioenia,’ 18,
Ap., 1891), by the method of mixture between 0° and 30°, gave a curve very similar
to Rowland’s, but with a minimum at 20° C. This excessive lowering of the
minimum may probably be attributed to constant errors inherent in their methods of
experiment.

VOL. CXCIX. — A.

T

138

PROFESSOR HUGH L. CALLENDAR OX

The later work of Ludix (Zurich, 1895), under the direction of Professor Pernet,
extending from 0° to 100° C., requires further notice on account of the wider range of
the experiments, and the great attention paid to the thermometry. He employed
mercurial thermometers of the Paris type, with all the usual precautions. He adopted
the method of mixture, adjusting the quantity of hot water in each case to give the
same rise of temperature, from 11° to 18°, in the calorimeter. The discrepancies of
individual measurements at any one point do not exceed '3 per cent., but he did not
vary the conditions of experiment materially, and it may be questioned whether the
well-known constant errors of the method could have been eliminated by the devices
which he adopted. His results (see Table XIV., p. 144) give a minimum at 25° and
a maximum at 87° C., the values being ‘9935 and P0075 respectively, in terms of
the mean specific heat between 0° and 100°. The rapid rise from 25° to 75° may
possibly have been due to radiation error from the hot water supply ; and the
subsequent fall between 90° and 100° to the inevitable loss of heat by evaporation of
the nearly boiling water on its way to the calorimeter. His values, reduced to a unit
at 20° C., are given for comparison in Table XIV. The agreement with our values is
remarkably close at the lower temperatures ; but the difference at 80° amounts to
nearly 1 per cent. In addition to the fact that his curve cannot easily be reconciled
with Regnault, there is this theoretical difficulty in accepting his values at higher
temperatures. The quantity which he actually observed was the mean specific heat
between the higher temperature and the final temperature of his calorimeter. His
values of the specific heat itself were obtained by differentiating the curve, and really
depend on small differences at the higher points between observations which are
themselves difficult and uncertain. His values for the mean specific heat differ much
less from ours. The peculiar advantage of the method we adopted is that the specific
heat itself is determined over a range of 8° to 10° at each point by adding accurately
measured quantities of energy to the water at the desired temperature. There is no
possibility of evaporation or heat-loss in transference as in the method of mixture, and
the protection from external radiation is much more perfect.

(44.) The Work of Miculescu.

The work of Miculescu (‘Ann. Chim. Rhys.,’ XXVII., p. 202, 1892), though not
directly affecting the question of the variation of the specific heat, is interesting as an
example of a steady-flow method of calorimetry. The apparatus consisted of a Joule
calorimeter mounted horizontally, with the paddles directly driven by an electric
motor, the torque being observed by the “ cradle-method ” of Despretz and Brackett.
The heat generated was measured by passing a steady current of water round the
calorimeter, and observing with a platinum-iron thermocouple the rise of temperature
of the stream. The rate of heating was about 50 calories per second, the rise of
temperature 2°, and the capacity of the calorimeter about 3 litres. The work has

CONTINUOUS ELECTKIC CALORIMETRY.

139

been criticised by Schuster, and also by Ames (‘ Paris Congress Reports,’ 1900) on
the ground that the length of the lever by which the torque was measured was
apparently taken as 28 centims. without verification. M. Guillaume replies in a
footnote that the length would probably in anjr case be accurate to a tenth of a millim.,
or 1 in 2800 (which is about the limit of accuracy claimed for the best absolute
measurements of the mechanical equivalent), but regrets that the original lever
cannot be found for the purpose of verification. It appears, however, from the
description given by Miculescu, and also from the wood-cut of the apparatus, that
the weight was not supported on a knife-edge but suspended from a round liook
clamped to the lever with a screw, so that it would be a matter of some difficulty to
estimate the effective length. Moreover, it would be very difficult with a platinum-
iron couple to measure so small a difference as 2° with a degree of accuracy higher
than -01°, and Miculescu does not mention some of the necessary precautions. In
addition, the correction for the external loss of heat from the calorimeter by radiation
and conduction was regarded as being negligible in comparison with the heat- supply,
and no correction was applied for it. It is easy to estimate, however, from the
dimensions of the apparatus, that it could not have been much less than 0'5 per cent.,
and that it was probably larger on account of the smallness of the interval between
the calorimeter and the jacket, which would considerably increase the heat-loss by
conduction and convection through the air space. There are several other indications
in the paper that the author did not really aim at a higher order of accuracy than
1 per cent., which is about the limit usually reached in engineering experiments, and
that the work should be regarded rather as an illustration of a method than as a
serious absolute determination.

(45.) The Work of Reynolds and Moorby.

The work of Reynolds and Moorby (‘ Phil. Trans.,’ A, 1897), on the mean specific
heat between 0° and 100° C. in absolute measure, stands in the same category as that
of Rowland as an accurate determination of the mechanical equivalent. The two
are not directly comparable, but if we assume the rate of variation of specific heat
found in our experiments as the medium of comparison, the result of Reynolds and
Moorby would give 4T79 for the number of joules in one calorie at 20° C., at which
point Rowland’s corrected results give 4T81. If we took Ludin’s table, we should
find 4T57. The formula of Winkelmann* gives 4T33, and that of Regnault 4TG7.
The comparison of Reynolds and Moorby with Rowland is therefore a confirmation
to some extent of the accuracy of the present experiments, as compared with
Ludin’s, or with formulae proposed by other writers. On the strength of this
comparison, I have generally adopted the mean value 4 ‘180 joules per calorie at
20° C. as the most probable estimate of the “ Mechanical Equivalent,” assuming that
the absolute value cannot be certain to a higher order of accuracy than 1 in 2000.

* Winkelmann, ‘ Handbuch der Physik,’ Band 2, Abth. 2, p. 338.

T 2

140

PROFESSOR HUGH L. CALLEXDAR OX

It might, however, be claimed that either or both of the absolute determinations
above cited were more accurate than this, and that the error lay with our comparison.
It is a nice question whether the mean specific heat, which is practically indejiendent
of the scale of the thermometer employed, should be chosen as the standard of
reference, rather than the specific heat at a particular temperature such as 20° C. on
the scale of a particular thermometer. My own opinion is that the latter unit is the
more practical, and the most accurately reproducible. It is not very difficult in a
thermochemical experiment to measure a quantity of heat to 1 in 1000 with a
suitable rise of temperature and a good thermometer with known scale errors. But
it is quite impossible to realize the mean thermal unit to this order of accuracy without
the most elaborate apparatus on a scale approaching that adopted by Reynolds and
Moorby.

The fundamental difficulty in the determination of the mean specific heat between
0° and 100° C. lies in the great range of temperature to be covered, and the
consequent risk of excessive or uncertain loss of heat. Reynolds and Moorby
endeavoured to meet this objection by working on a very large scale, and succeeded
in reducing the heat-loss to 5 per cent., even without the use of lagging. But in
working on this scale they encountered peculiar difficulties, which were not overcome
without great pains and ingenuity, and which must in any case have materially
affected the order of accuracy attainable by their method. Owing to the employment
of a steam-engine as motor, it was difficult to secure a high degree of steadiness in
the conditions of running, and the outflow-thermometer could not be read more
closely than a tenth of a degree owing to its incessant oscillations. The largest
variation recorded in the two trials of which full details are given, was 4 ’9° F. in
two minutes on the outflow temperature, and four or five revolutions per minute in
300 on the speed. The greater part of these variations being accidental would
disappear in the mean, but it is probable that there would be systematic errors of the
same order as the limit of accuracy of the instantaneous readings. In using so large
a quantity of water, it was impracticable to deprive it entirely of air, which caused
considerable trouble, owing to loss of heat by generation of steam in the air bubbles.
Again it was naturally impossible to enclose the brake in a vacuum-jacket, or shield
it from draughts and extraneous disturbances by means of an isothermal enclosure.
The greatest uncertainty appears to have arisen from damp in the lagging, which
caused the rejection of a number of trials.. The correction for conduction along the
4-inch shaft was of much smaller magnitude, though probably more uncertain on
account of the impossibility of determining the actual gradient in the shaft itself.
The extreme limits of variation of the recorded results were from 776'63 to 779’46
foot-pounds, but considering that the main sources of error above mentioned were
accidental, it is probable that the accuracy of the mean would be of a higher order
than might be inferred from the separate experiments.

CONTINUOUS ELECTRIC CALORIMETRY.

141

(46.) Empirical Formula.

The usual method of representing the results of a series of observations like the
present is to adopt an empirical formula of the type,

s = 1 -f- at bt~ + cfi -f dd + &c., . (3),

and to calculate the values of the coefficients a, b, c, d, &c., by the method of least
squares. This was, in fact, the method adopted by Ludin. It has the advantage of
providing a simple and ready rule, which is very generally recognized and applied ;
but it appears to me that it is in reality liable to several objections. Too much
weight is given to the observations at higher temperatures, which are necessarily less
accurate than the rest. The results obtained are in a great measure dependent on
the particular type of formula assumed, which is frequently inadequate to represent
the phenomenon, and is generally quite unsuitable for extrapolation. Moreover, the
method gives a fictitious appearance of completeness and accuracy, which is quite
misleading, as the calculated values of the probable errors contain no reference to
possible sources of constant error. It also generally happens that the terms of the
empirical formula are large and of alternate sign, so that the small variation required
is given as the difference between large quantities, which must be calculated to
several figures in applying the formula. The following formula of Ludin supplies a
good illustration of some of these points : —

s = 1 - •000766Gffi + -000019598^ - -000,000,1162^

± -0000025 ± -000004 ± ’000,000,03

The probable errors of the several coefficients, as calculated by Ludin, are given in
the second line below the coefficients to which they apply. The value of the specific
heat at 100° C. on Ludin’s formula is made up as follows : —

s - 1 - -076668 ± ‘00025

+ -19598 ± -040

- -1162 ± -030

1 + -0031

It is at once obvious that a formula of this type is quite unsuitable for the
representation of the variation of the specific heat over the whole range 0° to 100° C.
Moreover, since the maximum divergence of the specific heat from its mean value
over the range 10° to 70° is only 2 parts in 1000, according to the present series of

142

PROFESSOR HUGH L. CALLENDAR ON

experiments, it is evidently desirable from a practical point of view to employ the
simplest possible formulae for its representation. The variation from unity over this
range cannot be determined more closely than 2 or 3 parts in 10,000 (i.e., 10 per
cent, of the variation itself), so that it would appear ridiculous to employ coefficients
with five significant figures in the formulae.

For the above reasons, in the ‘British Association Report, ’ 1899, the following
simple formula was given as representing the results between 10° and 60° within the
limits of accuracy of the observations, in terms of a unit at 20° C. —

s = -9982 -f -000,0045 (t — 40)2 . (5).

It was stated that the variation of the specific heat near the freezing-point was
apparently more rapid than the formula indicated, and could he approximately
represented by the addition of the constant quantity '020 calorie to the total heat.
This correction to the formula has since been further verified, and may be represented
by the addition of a small cubic term below 20°.

Below 20° C. add to (5) -f- -000,000,5 (20 — ?)3 .... (6).

At that time the observations had not been extended above 60°, and the formula
of Regnatjlt, emended as already described, was therefore adopted for the higher
temperatures, namely,

Above 60° to 200° C. 5 = '9944 + -000, 04£ + -000, 000, 9£2 . . . (7).

Shortly afterwards, Dr. Barnes succeeded in obtaining six observations at higher
temperatures. One of these was vitiated by dissolved air, and another was incomplete.
There remain four good observations, which could be represented within 1 part in
10,000 by the linear formula,

From 68° to 92°. s = 1 + '000,14 (t - 60) . . . . (8).

This formula gives a value nearly 1 part in 1000 lower than (7) at 90°, and it
cannot be satisfactorily fitted on to Regnault’s observations at higher temperatures.
1 think on the whole it would be better to retain Regnault’s formula as previously
emended, until further observations are available. Although the agreement of the
four observations is so perfect among themselves, it is possible that they may be
affected by a constant error of this order of magnitude, if all the difficulties of the
work are rightly considered. Besides, the linear formula cannot represent the
probable increase in the rate of variation of the specific heat at higher temperatures,
which is theoretically required to account for the vanishing of the latent heat at
360° C., the critical temperature.

It would of course be easy to represent the observations a little more accurately in
any particular part of the curve by using more complicated formulae, but it is

CONTINUOUS ELECTRIC CALORIMETRY.

143

doubtful whether the advantage gained would be worth the extra complication, and
the possible confusion caused by changing a formula already published and widely
distributed.

Dr. Barnes (‘ Roy. Soc. Proc.,’ vol. 67, p. 242) has proposed the following : —

From 5° to 37°‘5,

5 = -99733 -f -000,0035 (37-5 - tf + ‘000,000,10 (37’5 - if . . (9).

From 37°'5 to 55°,

.s- = -99733 + -000,0035 (37'5 — tf — -000,000,10 (37'5 - tf . . (10).

From 55° to 100°,

6- = "99850 + -000,120 (t — 55) -f ‘000,000,25 (t — 55)2 . . . . (11).

The first two formulae differ only in the sign of the third term. They are not
quite so simple and convenient for calculation as (5) above. The first formula does
not represent the observations below 10 quite so accurately as formula (6), but both
are probably within the possible limits of error. The mean divergence of the
observations between 37°‘55 and 5° from the second formula is about 3 parts in
10,000, and is rather greater than the mean divergence of the observations from the
old formula (5). The agreement with the latter would be greatly improved if we

reject the discordant observation No. XXII., in series 2 at 54°"61 (Barnes, p. 244),

giving greater weight to the later observations, No. XXXII. in series 3 at 54°"57,
and No. XLVIII., in series 6 at 51°"02. On the whole, the balance of probability
appears to me to be in favour of retaining the older and simpler formula in preference
to that since proposed by Dr. Barnes.

The third of the above formulae, from 55° to 100°, does not fit with the second at
55°, the respective values being, ( L 0 ) "99893, and (11) '99850. This is shown by the
difference in Table XIY. between the values at 55° and 60° in the column headed
“Barnes, Roy. Soc., 1900,” which is only "00016. The corresponding differences for
five degrees on either side are 50° to 55°, "00087 ; 60° to 65°, "00061. There is also
a considerable change of slope from "000216 to "000120 at 55°. It is, of course, necessary
that the values should fit accurately at the point where the formulae meet, and it is
further desirable that there should not be a sudden change of slope. The first
condition is accurately and the second very approximately satisfied by the old
formulae (5) and (7) at 60°.

The following table contains a comparison of the formula of Ludjn and those of
Barnes, with those previously published in the B.A. Report, 1899.

144

PROFESSOR HUGH L. CALLENDAR OX

Table XIV. — Comparison of Formulae.

Temperature,

Centigrade.

Ludin, 1895.

Barnes,

Roy. Soc., 1900.

B.A. Report,
1899.

Values of J.

o

0

1-0084

1-0084

1-0094

4-219

5

1-0051

1-0055

1-0054

4-203

10

1-0026

1-0031

1-0027

4-191

15

1-0009

1-0012

1-0011

4-185

20

1-0000

1-0000

1-0000

4-180

25

0-9998

0-9991

0-9992

4-177

30

0-9999

0-9986

0-9987

4-175

35

1-0006

0-9984

0-9983

4-173

40

1-0017

0-9984

0-9982

4-173

45

1-0030

0-9986

0 • 9983

4-173

50

1-0046

0-9991

0-9987

4- 175

55

1-0063

0-9999

0-9992

4-177

60

1-0079

1-0001

1-0000

4-180

65

1-0094

1-0006

1-0008

4-183

70

1-0109

1-0013

1-0016

4-187

75

1-0123

1-0020

1 -0024

4-190

80

1-0131

1-0027

1-0033

4-194

85

1-0137

1-0034

1-0043

4-198

90

1-0136

1-0041

1-0053

4-202

95

1-0129

1-0048

1-0063

4-206

100

1-0117

1-0055

1-0074

4-211

For the reasons already stated, the values in the above table are all expressed in
terms of a unit at 20° C. for comparison, and are given to 1 part in 10,000 only. I am
inclined to regard the values and formulae corresponding to the column headed “ B.A.
Report, 1899,” as being the most probable. The corresponding values of J, the
number of joules per calorie, are calculated assuming the value 4T80 at 20° C.

(47.) Theoretical Discussion of the Variation of the Specific Heat.

Clausius (‘ Mechanical Theory of Heat,’ p. 180, translation, 1879) has calculated
the specific heat of water at constant volume C„ from that at constant pressure Cp
by means of the well-known equation,

C„ = C, — T (dv/dT)p {dp/tin, = C , - T(dr/dT)f/(dv/dp)v . . (12).

Taking Regnault’s values of Cy„ Kopp’s values of the coefficients ol expansion,
and Grassi’s values of the compressibility, Clausius finds the values,

Temperature . 0° 25° 50°

C„, at constant pressure . . 1'0000 l'OOlfi 1’0042

C,, at constant volume . . . (P9995 (P9918 (P9684

CONTINUOUS ELECTRIC CALORIMETRY.

145

Dieterici (‘ Wiedemann’s Annalen,’ vol. 57, p. 333) has repeated the calculation,
employing Rowland’s values of the specific heat at constant pressure. He seems to
argue that the variation of the specific heat at constant pressure as discovered by
Rowland is of the character to be expected from the large variation of the specific
heat at constant volume. I do. not think, however, that we can fairly infer anything
with regard to the variation of the specific heat at constant pressure from that at
constant volume. The value of the latter can only be deduced by the aid of very
uncertain data, and we have as yet no sure theoretical guide as to the way in which
it ought to vary, apart from experiments on the specific heat at constant pressure.

The most remarkable facts about the specific heat of water are its great constancy
over a considerable range, and its high value as compared with that of the solid or
the vapour. The specific heat of liquid mercury is nearly the same as that of the
solid, and both are about double that of the vapour at constant volume. The
specific heat of water is double that of ice, and nearly three times that of steam at
constant volume. If we adopt Rankine’s hypothesis of a constant “ absolute ”
specific heat for each kind of matter, we must admit that this absolute specific
heat has a different value in different states.

J. Macfarlane Gray (£ Proc. Inst. Mech. Eng.,’ 1889), adopting Rankine’s idea
of a constant specific heat for “ideal” water, gives, without proof, the following
formula for the total heat h of water reckoned from 0° 0.,

h = P0106 (6 - 273) + 0v dp/dd . (13),

in which the constant is the ideal specific heat, and dp/dd is the rate of increase of
the steam-pressure with temperature. The term 6v dp/dd is evidently intended to
represent the latent heat of expansion of the liquid from the ideal state against the
steam-pressure. The value of v, however, is not the apparent expansion from 0°C.,
but is the “ actual volume of unit mass of water less its absolute matter-volume, the
pressure during the heating being that due to the higher temperature. Absolute
matter is no doubt much more dense than platinum ; and the reduction from the
apparent volume, being very small, may therefore be disregarded.” The value of v
is therefore taken as the observed volume at the temperature 6 of unit mass of
water. The justification of this argument is not very clear, but the values
calculated by the formula on this assumption agree fairly with the observations of
Regnault on the specific heat between 110° and 190° C.#

W. Sutherland, in a recent paper “ On the Molecular Constitution of Water,”
(‘Phil. Mag.,’ vol. 50, p. 460, 1900), has endeavoured to explain the properties of
water on the assumption that it is a mixture of two kinds of molecules in varying
proportions, “ trihydrol,” 3HoO, which is identical with -ice, and “ dihydrol,” 2H.,0,
which constitutes the greater part of liquid water at higher temperatures. He

* In a recent paper (‘Proc. Inst. Civil Engn.,’ vol. 147, 1902), Gkay gives a further elucidation of this
formula, and a detailed comparison with our experimental results . — Added March 11, 1902.

VOL. CXCIX. — A.

U

146

PROFESSOR HUGH L. CALLEXDAR ON

calculates the densities and proportions of the two ingredients on the ' assumption
that each constituent follows the Mendeleeff law of expansion, or that the density
is a linear function of the temperature. He thus finds that 1 gramme of water at
0° C. consists of liquid 3HoO, density ’88, ’375 gramme ; liquid 2H.:0, density 1‘09,
’625 gramme. The latent heat of fusion of trihydrol, 16 calories, is calculated on
the assumption that it would expand ’0366 on fusing, wdiich is the mean expansion
on fusion deduced from a number of metals. The remainder of the latent heat of
fusion of ice is supposed to be made up of the heat ot dissociation of ‘625 trihydrol
into dihydrol, and the solution of the remaining fraction -375 in the dihydrol. As
a starting-point for the explanation of the variation of the specific heat, he assumes
that the specific heat of dihydrol is l’OO at 200° C., at which temperature water
contains ’167 of trihydrol. Taking “the usual rate of variation of the specific heat
of a liquid as T per cent, per degree,” he finds -83 for the specific heat of pure
dihydrol at 0° C. He takes that of trihydrol to be '6 with a similar rate of increase,
and explains the remainder of the specific heat of water as due to the heat of
dissociation of trihydrol and of solution in dihydrol, which he calculates on the
above assumed values of the specific heats.

It is more natural to regard the high specific heat of water as due to internal work
done against molecular forces, and as being closely related to the decrease of the
latent heat of vaporization with rise of temperature. Whatever assumptions are
made with regard to the molecular constitution of water, it was proved by Rankest e
(in a slightly different form), in 1849, that the rate of decrease of the latent heat
dhfdd was equal to the difference between the specific heat of water sK, and that of
steam S* at constant pressure.

dh/dd = S s — sK . (14).

This is accurately and necessarily true if we assume that steam may be regarded
as an ideal gas of constant specific heat, which is probably justifiable at low pressures.
At higher pressures, it is necessary to make allowance for the co-aggregation of the
steam molecules, which may be effected to a high degree of approximation by the
method which I have explained (‘ Roy. Soc. Proc.,’ Nov., 1900). If we write h for
the total heat of water from 0° C. (without assuming the specific heat to be constant),
we obtain the relation,

L - L0 = S0 (0 — 6»0) - (n + 1 ) {cp - c0p0) - h . . . . (15),

where S0 is the limiting value of S* at zero pressure, and c represents the defect of
the actual volume of the steam from the ideal volume R 6/p. This defect of volume
is independent of the pressure p, but varies inversely as the nth power of the
absolute temperature 6. It is clear that the rate of diminution of the latent heat is

CONTINUOUS ELECTRIC CALORIMETRY.

147

not constant, but increases at higher temperatures, on account of the co-aggregation
of the molecules, as represented by the increase of the product cp , where j) is the
saturation-pressure.

Similarly, we might suppose that the rate of variation of the total heat h of the
liquid would be constant for ideal water, hut increases at higher temperatures for
actual water on account of the existence of molecules of vapour in the liquid. The
two phases in equilibrium at saturation-pressure may be regarded as a saturated
solution of water in steam (the dissolved water being represented by the proportion
of co-aggregated molecules cp/^XO), the liquid phase conversely as a saturated solution
of steam in water. On the one hand, the total heat of the steam is reduced below
its ideal value by the amount ( n +1) cp, owing to the presence of dissolved molecules
of water causing a diminution of volume c per unit mass of the solution ; on the other
hand, the total heat of the liquid is increased by the presence of a certain proportion
of dissolved molecules of steam, which may doubtless account for part of the thermal
expansion of the liquid. When the temperature is raised, the properties of the two
phases continue to approach each other, as the proportion of water in the steam and
of steam in the water increases At the critical temperature the two solutions mix
in all proportions.

It is possible to estimate more or less perfectly the number of co-aggregated
molecules present in steam at any temperature by observing the defect of volume
from the ideal state ; or to deduce the value theoretically on certain assumptions
from experiments by the Jotjle-Thomson method on the cooling effect of expansion
through a porous plug or throttling aperture (‘ Roy. Soc. Proc.,’ 1900, vol. 67, p. 270).
It is probable that the proportion of steam molecules present in the liquid is similarly
related to its expansion, but there is no certain theoretical guide to the relation.
The simplest hypothesis to make would be that the number of vapour molecules per
unit volume of the liquid is the same as the number of molecules per unit volume of
the saturated vapour at the same temperature. If we suppose the formation of
vapour molecules in the interior of the liquid (specific volume w) to require the
addition to the liquid of the latent heat corresponding to the same quantity of
vapour (specific volume v) when formed outside the liquid (i.e., if we neglect the
heat of solution of the vapour in the liquid), the total heat h of the liquid would
require to be increased by an amount iv/(v — iv) of L to allow for the latent heat of
the dissolved steam. It happens that this result, though obtained by a quite different
line of reasoning, agrees with the expression given by Gray, and approximately
represents the experiments of Regnault at high temperatures. We thus obtain the
simple formula,

h — s()t -f wL/{y — iv) . (16).

On similar grounds it would be natural to suppose that the increase of the specific
heat, as we approach the freezing-point, was due to the presence of a certain

u 2

148

PROFESSOR HUGH L. CALLENDAR ON ELECTRIC CALORIMETRY.

proportion of dissolved ice molecules, which are also required to account for the
anomalous expansion. But here there is no obvious guide to the proportion required,
or to the latent heat of solution to he assumed. The basis adopted by Sutherland
appears to he too empirical, and the results to which it leads are too vague and
inexact to permit of a profitable comparison with experiment. In order to place any
such hypothesis as that represented by formula (16) on a sound experimental basis,
it would be necessary to determine the variation of specific heat of several typical
liquids with great accuracy over a wide range. The required data are not at present
obtainable, but the suggestion appears to be worth recording as a possible physical
interpretation of the- variation of specific heat.

INDEX SLIP.

Babnes, Howard Turner— On tlie Capacity for Heat of Water between the
Freezing and Boiling Points, together with a Determination of the
Mechanical Equivalent of Heat in Terms of the International Electrical
Units. Phil. Trans., A, vol. 199, 1902, pp. 149-263.

Calorimetry — Continuous Electrical Method.

Babnes, Howard Turner. Phil. Trans., 4, voL 199, 1902, pp. 149-263.

Heat — Mechanical Equivalent, Comparison of Mechanical and Electrical
Methods.

Babnes, Howard Turner. Phil. Trans., A, vol. 199, 1902, pp. 149-263.

Specific Heat of Water — Variation determined by Electrical Method.

Baenes, Howard Turner. Phil. Trans., A, vol. 199, 1902, pp. 149-263.

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[ 1«> ]

III. On the Capacity for Heat of Water between the Freezing and Boiling-Points ,
together with a Determination of the Mechanical Equivalent of Ileat in
Terms of the International Electrical Units. — Experiments by the Con¬
tinuous-Flow Method of Calorimetry , performed in the Macdonald Physical
Laboratory of McGill University , Montreal.

By Howard Turner Barnes, M.A.Sc., I). Sc., Joule Student.

Communicated by Professor H. L. Callendar, F.P.S.

Received June 15, — Read June 21, 1900.

Section

Table of Contents.

1. Introduction . ......

2. General theory of the method of Continuous Calorimetry .

3. Measurement of Fundamental Constants .

a. Clark cell ; b. Resistance ; c. Thermometry ; d. Time ; e. Weight.

4. Description of Apparatus and Method of making the Experiments .

5. Experimental Proof of the Theory of the Method .

6. Effect of Stream-Line Motion .

7. Preliminary Measurements of the Mechanical Ecjuivalent .

8. Determinations between 0 and 100 C. at different Temperatures .

9. The Variation Curve of the Specific Heat of Water in its relation to the work of other

observers .

Page

149

152

158

203

225

234

237

238

257

Sec. 1. — Introduction.

The unsatisfactory state of our knowledge of the Mechanical Equivalent of Heat and,
inseparably connected therewith, of the capacity for heat of water, is the more
surprising when we consider the large number of physicists who have devoted their
attention to this subject during the century just closed. Since the remarkable
pioneer experiments of Count Rumford, undertaken just 100 years ago, to determine
the nature of heat, the subject has been advanced step by step by different investi¬
gators. Conspicuous among these we may mention Regnault, who gave us the first
idea of the mode of the variation of the specific heat of water with temperature,
without, however, giving us any knowledge of the mechanical equivalent of heat ;
Joule, who gave us the first measurements of the mechanical equivalent without
attempting to study the thermal unit at different temperatures; Rowland, who by
the remarkable accuracy of his experiments gave us not only a direct determination
(314.) 13.8.02

150

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

of the mechanical equivalent, but also the variation of the thermal unit over a
limited range. More recently we have the exceedingly careful experiments of
Miculescu, of Griffiths, of Schuster and Gannon, and of Reynolds and Moorby.

It is evident from only a cursory glance at the work of these and the host of other
investigators, that the science of calorimetry must he regarded as incomplete and
approximate so long as its fundamental unit remains in doubt. To obtain, as is
urgently needed, a complete series of determinations of the capacity for heat of water
over the entire range of temperature is manifestly impossible by the older methods of
calorimetry. A new method has long been required, more completely free from the
influence of extraneous surrounding conditions.

During a conversation which I had with Professor Callendar, in the autumn
of 1896, we discussed the unsatisfactory state of our knowledge of the specific heats
of water and mercury. Professor Callendar pointed out that what was required
was a new method of calorimetry, which would reduce to a minimum many of the
larger corrections inherent in, and making uncertain, the older methods. Such he
considered possible in a continuous, or steady, flow method, in which a stream
of liquid could be made to continuously carry oil' a definite and measurable supply of
heat. This method he considered capable of great accuracy and free from nearly
all the errors in the older methods. I very gladly consented to assist Professor
Callendar in developing this method, which we commenced as a joint work early
in 1897.*

The early experiments with mercury will be discussed in full in another paper, and
cannot be more than mentioned in this place. They were satisfactory in many
respects, but must be considered more as preliminary attempts, the experience of
which served so much to aid in later measurements with water. A calorimeter,
designed for the determination of the specific heat of water, was set up and tested
just previous to the meeting of the British Association at Toronto, to which body a
preliminary note was sent describing the method in general terms. On the re-opening
of the College session, in September of that year (1897), Professor Callendar was
unfortunately obliged, through stress of work, to relinquish his connection as a joint
observer in the experiments. My own duties, however, were such as to allow of a
certain amount of time to be devoted to research, so that the work was carried on at
intervals throughout the winter. During this session Mr. Russell W. Stovel, B.Sc. ,
joined our graduate classes and devoted a large part of his time to assisting in the
work. It is largely to his skill as an observer that it was possible to continue the
work during this time.

In the spring of 1898 Professor Callendar was called to London to fill the Quain
Chair of Physics in University College, and was obliged to sever his connection
entirely with the experiments. It was with extreme regret that we realized this, as

* For theory of experiment and work done prior to 1897, see the paper liy Professor Callendar
above, pp. 55-148.

BETWEEN THE FREEZING AND BOILING-POINTS.

151

so much was due to his kind supervision in perfecting different portions of the
apparatus. Mr. Stovel also was obliged to leave at this time, so that the work was
somewhat delayed.

During the summer of 1898, Mr. Charles Sheffield, B.Sc., was kind enough to
devote his entire time to the work, and made himself exceedingly useful through his
untiring efforts until late in the autumn. University duties being closed, it was
possible for us to devote all our time to the work. The measurement of the
mechanical equivalent we obtained will be described 'further on, but it must be
regarded as a preliminary attempt Owing to a source of error in the method, which
was not discovered until some time after. We made a careful study of the general
theory of the method, which, as will be explained in its place, was affected somewhat
by the error above mentioned. On comparing these determinations with later ones,
and more particularly in applying the theory of the method to different calorimeters,
we met with such large discrepancies, much larger than any possible error in the
instrumental readings, that we were forced to abandon the greater part of our earlier
results, and re-organize the experiment.

Unfortunately Mr. Sheffield was called away at this time and was unable to
continue his work on the method. It is with extreme regret that I have to record
the death of Mr. Sheffield, since leaving this laboratory, which occurred recently at
Niagara Falls, where he occupied a position in the capacity of electrical engineer.
His death at so early an age and under such trying circumstances is all the more sad,
as he had proved his worth and ability in so many ways as an accurate observer and
faithful worker.

During the winter of 1898-99 I was obliged to undertake the sole responsibility

s

of the work, with the exception of some temporary assistance in taking observations
from my colleague, Mr. H. M. Tory, M.A., to whom I am also indebted for many
helpful suggestions. As at that time there was no prospect, until the close of the
session, of finding an experienced observer who could devote sufficient time to help
in taking observations, it was necessary for me to arrange the. conditions of the
experiment so as to be able to take all the observations, both thermal and electrical,
myself. With a little practice I was soon satisfied that this could be done, although
not quite so quickly as with two observers, yet with sufficient accuracy to satisfy the
conditions of the experiment. It became chiefly necessary to produce perfectly
steady and uniform conditions, over a more extended period of time, conditions which
demanded greater refinements in the apparatus. The experiments from this time on,
as they became little by little improved and extended, were so steady and consistent,
and fulfilled the conditions demanded by the theory so perfectly, that it was deemed
unnecessary to break the continuity of the work by introducing a second observer.
From January, 1899, to the close of the work, the complete set of observations for
nearly every experiment was taken by one observer.

The results of the work from 4° C. to 60° C., obtained between January and June

152

DE. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

of that year, were communicated by Professor Callendar and myself to the meeting
of the British Association, at Dover, in September. A reprint of this communication,
slightly modified, to contain some later determinations above 60°, was published in
the ‘ Physical Review’ of April, 1900.

In the present communication 1 desire to record the complete set of experiments
obtained for the mode of variation of the specific heat of water over the entire
range 0° C. to 100° C., feeling confident that they represent, to an order of accuracy
approaching 1 in 10,000, the true values, and to point out the wonderful verification
they give of the work of Regnault over the range where his experiments are the
most trustworthy, a verification so complete that the present work may be said
to extend over the entire range where it is possible to maintain water in the
liquid phase.

I desire, at this time, to record my thanks to Professor John Cox, Director of the
Macdonald Physical Laboratory, for placing every facility at my disposal that could
aid me in the work ; to Mr. J. W. Fraser, B.Sc., Demonstrator in Physics in this
laboratory, for his observations on the comparisons of our 1-ohm resistance standards;
and to Mr. G. W. Scott for his kindness in helping me prepare figures for this paper.

I am also indebted to Messrs. Eimer and Amend, of New York, for the very
efficient way in which they made three glass calorimeters, and the great trouble they
took to exhaust very perfectly the vacuum-jacket connected with each one.

Sec. 2. — General Theory of the Method of Continuous Calorimetry .

If we have a flow of liquid, Q per second, continuously heated by an electric
current in a fine tube enclosed in a vacuum-jacket, the walls of which are maintained
at the temperature of the liquid flowing into the fine tube, then, when equilibrium
has been established,

Js Q t (6l — Of) -f (#i — ) ht = EC H,

where

J is the mechanical equivalent of heat,

s, the specific heat of the liquid,

Cj, the temperature of the inflowing liquid,

6-1, the temperature of the outflowing liquid,

h, the heat-loss per degree difference in temperature between the surface of
the fine tube and the walls of the vacuum-jacket,

EC, the electrical energy generated per second, and

t, the time of flow.

If the liquid be a conductor of electricity, such as mercury, then E represents the
difference of potential maintained across the column of liquid in the fine tube, and C
represents the current flowing through the tube. If the liquid be a non-conductor,

BETWEEN THE FREEZING AND BOILING-POINTS.

L53

such as water, then E represents the difference of potential across a conducting wire
passed through the tube in which the current C flows.

We will deal entirely in this place with the method as applied to this latter case,
such as a steady flow of water, and replace Js in the general equation by J, or the
number of joules in one calorie.

Let J = 4 ‘2 (1 8), where § is a small quantity varying with the thermal

capacity of the water, then we may write the general equation,

4-2 (1 ± §) Qt (01 - 0O) -f (01 - 0O) lit = EC*.

Dividing through by t, and re-arranging the terms, we have

4-2 Q {01 - 0Q) S + (0l - 0O) h = EC - 4-2 Q (0, - 60).

This we will call the general difference equation.

The total heat-loss from the water will be made up of radiation from the surface ot
the tube through which the water is flowing, conduction from the ends of the tubes
containing the thermometers for measuring the temperature of the inflowing and
outflowing water, and convection currents due to insufficient stirring around the
thermometer bulbs. There will be a small gain in heat due to work done by
the water in flowing through the fine tube. To study the effect of these upon
the general difference equation it will be necessary to refer to the diagram of the
continuous-flow electric calorimeter in its simplest form, given in fig. 1.

A

w

1P-

Fig. 1. Diagram of Calorimeter.

y l

In this AB represents the fine tube in which the water is heated while flowing-
through, V, the vacuum-jacket, and C and P the inflow and outflow tubes connected
to AB, in which the thermometers are placed. The water-jacket is shown at W, and
includes the vacuum-jacket and inflow-tube C. The water enters the calorimeter at
E from a reservoir separate from that supplying the water for the jacket, but
maintained at the same temperature. The electric heating current passes through
the fine tube AB through a platinum wire extending the whole length, but is
arranged so as not to generate heat in the vicinity of the thermometer bulbs. The
thermometer in C measures the temperature 60 of the inflowing water, and that in P
VOL. CXCIX. — A.

X

154

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

the temperature 01 of the outflowing water, warmed by the passage of the electric
current. The temperature of the water increases rapidly from B to A, and gradually
decreases from A towards P.

Radiation. — The loss of heat through the vacuum-jacket will consist of the
cooling of the surface of the glass in the flow-tube by radiation, by convection
currents of residual vapour in the jacket, and by radiation from the molecules of the
water itself. Provided the vacuum is good, these are all included in the h term of
the general difference equation. This radiation term obviously should not vary, hut
should remain independent of the quantity of water flowing, provided it can be
assumed that the temperature gradient from B to A remains the same for all flows.
This assumption can be justified only if the temperature gradient is linear. If it is
not linear then we may have either one of two conditions : —

1. When the distribution of heat in the water column AB is such that the
water is hotter in the centre than the sides, in which case the temperature
of the glass surface of the fine tube will be that represented in (1) fig. 2.
and will depend on the thermal conductivity of the different layers of

water between the centre and the sides, which will be conditioned by the
rapidity of flow. This condition is fulfilled perfectly when the water
column is receiving heat from a central wire conductor and flowing at
velocities less than the critical velocity for the tube in question. In this
case the water flows in parallel stream-lines, and does not mix. The
higher the velocity of flow up to the critical velocity, the more gradual will
be the slope of the temperature gradient of the glass surface from B to A.
At A, the water is mixed around the thermometer bulb and the temperature
of the glass suddenly increases. For any given temperature 6X of the
water, as indicated by the outflow-thermometer, the total heat-loss from
the water will decrease with increasing velocity of flow in proportion to
the slope of the temperature gradient along BA.

2. The case where the sides of the water column are hotter than the interior, or
where the water is receiving heat from the surface of the fine tube. The

BETWEEN THE FREEZING AND BOILING-POINTS.

155

temperature gradient will then be represented by (2) fig. 2. The water
flowing through the interior of the tube will receive less and less heat from
the layers along the sides as the velocity of flow increases. To attain the
given temperature 0] on the outflow-thermometer, the temperature gradient
from B to A will rise rapidly, and suddenly decrease as the water is mixed
around the bulb. The total heatdoss from the water will then increase
with increasing flow in proportion to the slope of the temperature-gradient
from B to A. This condition would be perfectly fulfilled by replacing the
fine glass tube by one of metal through which the electric heating current
could be made to flow. Less perfectly it is fulfilled when the heating wire
lies along the sides of the tubes, and supplies heat to the layers of water
nearest the walls of the tube.

In both these cases the heat-loss would not be independent of, hut would depend
on, some function of the flow. To ensure a perfectly uniform temperature, equal to
that of the flow-tube, throughout any section of the water column, it is necessary to
produce thorough mixing at all points, and avoid the formation of stream-lines. If
this is fulfilled, we can be safe in assuming the temperature gradient at least
approximately linear from B to A in both the above cases. Also that the total
quantity of heat lost per second by radiation from the water in its passage through
the length of tube included in the vacuum-jacket is the same, quite independent of the
velocity of flow.

An experimental study of the two cases above given, where the water flowing in
parallel stream-lines receives heat from a platinum wire, which may be moved from
the centre to the sides of a 3-millim. bore flow-tube, will be given in Section 7.

Conduction. — The heat-loss by conduction from the ends of the calorimeter will
evidently be very small where a bad thermal conductor, such as water, is used.
Where metal wires are introduced to convey the electric current to the central
heating wire, the conduction of heat from the water by the wires assumes a much
more serious character, more especially when the calorimeter and jacket are maintained
at a temperature very different to that of the surrounding air. At the inflow end
the effect, when the calorimeter is at a higher temperature than that of the air, is to
lower the temperature 0O of the inflow water by a small amount. The effect at the
outflow end is similar, but smaller, on account of the direction of flow. It is evident
that this can always be measured and eliminated for any given flow by recording the
temperatures of the inflow and outflow thermometers before the electric* heating
current is turned on. The only conduction effect that these “ cold ” readings will
not take account of is the conduction from the outflow-tube due to the rise of
temperature (01 — 60). This must be separately measured, in other ways. It can be
estimated and its maximum effect obtained, for any given difference in temperature,
by surrounding the outflow-tube beyond the water-jacket by a water circulation, the
temperature of which can he changed at will. If it is made as small as possible by

x 2

156

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

replacing the water circulation by heavy lagging ; its> effect can be measured by
varying the flow of water, as will be shown further on. This conduction effect will
be independent of the difference in temperature between the jacket water and the
outside air, and depend on the rise of temperature (d, — d0 ) directly, and on the
velocity of flow inversely.

Convection. — We have already discussed the effect of the stream-line flow on the
radiation correction, when the water is not stirred in its passage through the flow-tube.
It is proposed further on to treat this more in detail, as it has an important bearing
on the general validity of this method applied to a non-conducting liquid. The effect
of convection currents around the thermometer bulbs is avoided by suitably stirring
the water. Strictly speaking, the thermal stream-lines in the flow-tube should not
be classed as convection currents, but I have included them here for the sake of
convenience.

Gain of Heat. — The work done by the water in flowing through the fine tube may
be measured by determining the difference in water-pressure between the inflow and
outflow-tubes, for any given flow. The work done by any other flow can then be
determined by measuring the change in temperature on the outflow-thermometer due
to the change in flow.

If W, be the work done by the flow Qx per second,

’’2 j ) 5 > ) ■> Q$ i >

then

Wj = JQX (61 — d0) + (6l — d0) h and W2 = JQ3 (d3 - d0) + (d2 - d0) A,

where d0 is the temperature of the inflowing water as before,

0l and the temperatures of the outflowing water for the flows Q, and Q.:
respectively, and

h the heat-loss per degree rise, as before.

Then W1 - Wa = J (Qj - Qo) (01 - d2) + h ( d ! - do).

But h is small, and (0l — do) is small, so that we may neglect it in comparison to
(Qi — Qo). If we find that (dx — - do) is negligible for a large value of (QL — Qo), then
we have W, = W3 = 0.

For the limits of flow and the size of flow-tubes I adopted in the present experi¬
ments, no measurable effect could be obtained on the outflow-thermometer. Even if
the work done was appreciable, the method adopted of obtaining the “ cold ” readings
for each flow would eliminate it, except if it varied with the change in the viscosity
of the water, heated through the temperature (dj — d0).

Method of Measuring the Specific Heat. — "Referring to the original difference-
equation, we see that

4-2 Q (01 - d0) 3 + (01 - d0) h = EC - 4-2 Q (dx - d0),
in which there are the two terms 3 and h to be determined. If wre take two flows of

BETWEEN THE FREEZING AND BOILING-POINTS.

157

water, Qt and Qo, for the same inflow temperature d0, then we have the two
equations

4-2 Q, (0t - e„) S, + (tf, - e0) h = EjC, - 4-2 Q, ( 6 \ - 0O),

4'2 Qo (d2 — d0) So + (do — 90) h = E2C3 — 4-2 Q2 (do — d0).

If the electric current is adjusted for the two flows so that dt = do, then

(dx - d0) h = (0i— d0) h.

and Si = So

S, and hence by direct subtraction and writing t/d = (d1 — d0) = (do — d0),

4-2 (Qj - Qo) d0S = (ElCl - 4-2 Qj dd) - (E2C2 - 4'2Q3cZd),

and

S =

(EjC! - 4-2 Q, dd) - (E2C3 - 4-2 Q 3d0)
4-2 (Qx — Qo) dd ~ ~ ~ ’

from which J, or the number of joules per calorie = 4'2 (1 ^ §). By substituting S
in either difference-equation, h can be obtained.

The value of J thus obtained will be the mean over the range cl6 through which
the water is heated, and apply to the mean temperature

+ 2 (#1 - 0O) = T (mean).

If the variation of the value of J is not linear over this mean temperature, then for
different values of d0 and 9l for the same value of T (mean), the value of J will be
slightly different.

Application of the General Difference Equation to Test the Theory of the Method. —
In order to test the accuracy of the assumptions made in regard to

(a) The dependence of the heat-loss on the rise of temperature,

(b) The dependence of the heat-loss on the flow, including the conduction

correction,

we will consider the general difference equation. We have as before

4-2 Q (cW) S + (dd) h = EC — 4-2 Q (dO).

Dividing through by d9 , the equation is expressed per degree rise, or

4-2QS + A =

EC - 4-2 Q dd
dd

158

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

If h depends only on dd, then for different values of dd, for the same mean
temperature and flow, we have

4‘2 Q 8 fl- h — A = constant.

This relation should hold provided the temperature coefficient of both 8 and h is
linear. A small variation from lineality can, however, be safely neglected.

If we vary the flow and keep the rise of temperature constant, then we have in the
equation already given

4-2 Q 8 + h = A.

The value of A for different values of Q will vary in proportion to 4-2 Q 8, but the
variation will be a linear one, provided we are not neglecting any term on the left-
hand side varying inversely as the flow. If 8 = 0, then

7 EC - JQ dd

h = — w-

for any value of Q.

In Section 5 the experimental proof of these considerations is given, and it is
shown that within wide limits of flow it holds with great accuracy. For very small
values of Q the conduction becomes measurable, but the limits chosen in the present
series of experiments are seen to hold for the higher temperatures as perfectly as for
the lower.

Sec. 3. — Measurement of Fundamental Constants.

Owing to the importance attached to the measurement of the different constants
in the general equation of the method, it is proposed to treat each one separately in
this section, dividing them up under the two heads electrical and thermal. In the
first we have the Clark cell, standard resistance and potential measurements, and in
the second the measurement of temperature, weight and time.

The general plan of the electrical connections is given in fig. 3. A large 4-cell
accumulator, of 200 ampere-hours each, supplied the steady heating current to the
calorimeter through the resistance and rheostat. Potential terminals were taken
from the calorimeter and resistance, and from two Clark cells in series, to the paraffin
block, where they were placed in mercury cups cut in the solid paraffin. Wires
leading from two holes, placed equi-distant from the other cups in the block, were
carried to the potentiometer shown to the left, and included a galvanometer in the
circuit. By interchanging two connections, the Clark cell, calorimeter or resistance
could be connected through the galvanometer to the potentiometer. By altering the
rheostat or connecting a smaller number of cells, the heating current could be

BETWEEN THE FREEZING AND BOILING-POINTS.

159

adjusted for a change of water flow in the calorimeter, so as to produce the same rise
of temperature.

If X0 is the reading of the potentiometer for the balance point of the Clark cell,

Xx potentiometer reading for difference of potential on calorimeter,

X2 the same for resistance R,
e the E.M.F. of Clark cell,

R the value of the resistance,

then E, the potential across the calorimeter, is Xj/Xq X 2e, where the two Clark cells
are used in series.

Clark cells.

Also the current C in the circuit is X2/X0 X 2e/R, from which we get the total
watt energy per second supplied to the calorimeter, when the conditions have become
steady, and used in heating the water,

EC =

XjX/4

X02E ’

The experimental error involved in the measurement of EC will depend on the
accuracy of the measure of Xx, X2, and X0 and on the constancy of e and R.

Sec. 3 a. — Clark Cell.

Some time previous to my undertaking the present series of experiments, I made a
careful study of the Clark cell with Professor Callendar in order to become more
conversant with its behaviour, as well as to devise, if possible, a more reliable form of
cell than the one in vogue at that time. The result of this work has already been

160

DE. H. T. BAENES ON THE CAPACITY FOE HEAT OF WATEE

published in full (‘ Proc. Roy. Soc.,’ vol. 67, p. 117 (1897)), and in consequence may
be passed over here with but a brief mention. A thorough study of the old form of
cell recommended by the Board of Trade formed one of our chief objects, including
measurements of the diffusion lag on a sudden and definite change of temperature.
It was shown that for a change of 15° C. the time required for a B.O.T. cell to assume
its true value was of the order of 14 days when left undisturbed, but only 2 days
when shaken three times at different intervals. In cells where the saturated solution
of zinc sulphate was replaced by moist zinc sulphate crystals, no such effect could be
noticed, but the cells assumed their normal value on a sudden change in temperature
in 10 or 15 minutes, or, in other words, in only such time as was required for the cell
to assume the temperature of the surroundings.

These modified B.O.T. cells were studied in every detail, but more particularly as
regards the formula governing the variation of the E.M.F. with temperature and
their reproducibility. The remarkable constancy of these cells, their agreement
amongst themselves, and the closeness with which they followed the temperature
expression deduced, was a matter of much satisfaction. It was found that the
tenrperature change of the E.M.F. depended on two conditions, a change of
temperature and a change of strength of solution. These two changes were about
equal and formed one-half of the total change. By keeping the strength of the
solution constant and varying the temperature, the change was practically the same
for all strengths, and equal to that found by Professor Carhart for the Carhart-
Clark cell. If the temperature was kept constant and the strength of solution
varied, then the E.M.F. followed the concentration in the linear relation in millivolts
and grammes per cub. centim.,

c/E = 42-0 - 8 8 '0u>.

The variation with temperature was followed for higher temperatures, and the
transition point for the inversion of the heptahydrate (normal) crystals at 38 '78° C.
fixed. Various types of cells were devised which have been designated as the B.O.T.
“ crystal” cell, which is the modified Board of Trade form ; the “ sealed ” cell, which
is a form hermetically closed by glass fusion; and the “inverted’ cell, which is
a B.O.T. crystal cell reversed so as to place the negative electrode (zinc amalgam) at
the bottom. These all have an identical temperature formula, which may be expressed
in millivolts,

E, = E15 - P200 (t - 15°) - -0062 (t - 15°)3
between 0° and 30° for a mean temperature of 15°, or

E, = E39 — 1-6 3 5 (t — 39°) — -0140 (t — 39°)°
for a mean temperature of 39°.

BETWEEN THE FREEZING AND BOILING-POINTS.

161

For a cell about the mean temperature of 39°, with the hexahydrate crystals as
solid phase, the formula

E, = E39 - 1-000 (t - 39°) - -0070 (t - 39°)2

was obtained. For temperatures above 30°, as the second formula shows, the values
given by the first formula diverge from the observed values, due probably to
a secondary change produced by the decomposition of the mercurous sulphate. If
the first formula is corrected by the additional term

- -00006 (t - 150)3,

the calculated values from 30° to 40° C. are brought into very close agreement with
the observed values.

The Clark cells I have used in the present work are some of the original crystal
cells described in the paper by Professor Callendar and myself, “ On the Variation
of the Electromotive Force of the Clark Standard Cell,’’ already referred to. These
cells have been in the laboratory since 1895, and frequent comparisons made of their
E.M.F. with newer cells constructed at different times, both by myself and the
advanced electrical engineering students. As these cells are the originals from which
the temperature formulae already given were obtained, the constancy of the E.M.F.
maintained to the present time is a matter of some surprise, considering the severe
treatment they were subjected to during our earlier experiments. They were made
in the generally accepted way in a test-tube, and sealed by means of a cork, on the
top of which marine glue was melted. The life of such a cell is necessarily dependent
on the speed with which the crystals commence to dry, and this fact has been raised
against the use of moist crystals in place of a saturated solution. I have found
however, that in point of usefulness our crystal cells have outlived several cells with
saturated solution which were made at the same time. It appears that the crystals
retain the moisture more tenaciously than the saturated solution does, so that whereas
a solution may be reduced to one-third of its original bulk, with deposition of
crystals, a mass of crystals retains its moisture without diminishing in bulk or
uncovering the zinc rod. Owing to the dryness of the Montreal climate, the question
of the slow evaporation of liquid from the cells is a serious one. Our cells have been
re-sealed on one occasion by simply re-melting the marine glue, but apparently
without harm except to one (X3), which when left undisturbed for several months
returned to its original normal condition, and is at present as good as the others.

Several sealed cells, inverted cells, and a number of new crystal cells have since
been made in the laboratory, and have served to check the constancy of the original
crystal cells. Independently several cadmium cells were made in 1897, in the
inverted form, which proved to be quite satisfactory, and a comparison of the mean
of these cells was made with the mean of the crystal cells. These cadmium cells

VOL. cxcix. — A.

Y

162

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

I have described in another place, and have shown that they cannot be relied on, as
an accurate laboratory standard, to quite the same order of accuracy as a Clark
cell, although as a commercial instrument they have distinct and unquestionable
advantages over the Clark cell.

The method of keeping the Clark cells at a constant temperature has been already
described in my earlier papers. Briefly it consists of a water thermostat with gas
regulator, which is capable of maintaining the temperature constant to '02° C. over
extended periods. Whenever one of the experiments on the specific heat of water
was performed, the bath was set to regulate as near 15° as possible, and throughout
hardly ever varied more than '01° or '02°, unless some sudden change in the gas-
pressure or water supply introduced a disturbance of too sudden a nature to be at
once rectified by the regulator. The bath was supplied by a stream of water from a
constant-level head through a spiral of copper tubing about 2 millims. diameter, and
was heated by the gas flame, controlled by the regulator, as it passed through.
During the winter, the water-supply in the laboratory was always between 8C and
10° C. at the place where the bath was located, so that there was no difficulty in
maintaining the bath at 15b During the summer, however, the water sometimes
reached 18° or 20°, and it became impossible to keep the bath at 15° without running
the inflowing water through an ice tank before it entered the bath. As this entailed
considerable trouble, the bath was allowed simply to take the temperature of the
inflowing tap water, and rose and fell in temperature slowly with it. There was no
sj3ecial object after all in keeping the cells at 15°, on account of the accuracy of the
temperature coefficient, and the complete agreement of all the cells with one another
at all the temperatures of comparison. The temperature of the bath was taken with
a Geissler thermometer reading to ‘01°. This thermometer was reduced to the
nitrogen scale by comparisons, with a platinum thermometer, made both by Professor
Callendar and myself in 1896. It has seldom varied more than a few degrees
either way from 15° since then, and as it was a somewhat old thermometer at the
time of comparison with the platinum, it is unlikely that its readings have changed
much since. Moreover, our later tests on the temperature coefficient made with this
thermometer and thermometers calibrated by it, have agreed so well with the
earlier measurements that there is no reason to doubt the correctness of its readings.

The comparison of the E.M.F. of the different cells was made on a specially
constructed potentiometer, but as it has already been described it will be unnecessary
to more than mention it here. Special attention was given to having the readings
sufficiently sensitive to the order of accuracy we attempted, and defective insulation
was amply guarded against. For differences in E.M.F. the potentiometer read
directly in millivolts, at the rate of '01 mv. for each millimetre of scale. A
6000-ohm galvanometer in the circuit was sensitive to a scale distance a little less
than 1 millim.

In Table I. I have arranged the complete series of comparisons made on six of our

BETWEEN THE FREEZING AND BOILING-POINTS.

163

Table I. — Differences Expressed in Hundredths of a Millivolt from Mean.

1898.

March.

April.

9.

10.

11.

M.

16.

23.

25.

31.

2.

12.

20.

Xi .

- 5

- 6

- 4

- 4

- 5

- 4

- 4

- 4

- 5

- 4

- 4

X, .

+ 7

+ 8

+ 8

+ 7

+ 9

+ 7

+ 8

+ 8

+ 8

+ 8

+ 9

x3 .

x5 .

-17

-15

-13

-16

- 16

-16

-14

- 16

-17

-16

-16

X10 .

+ 14

+ 12

+ 10

+ 15

+ 15

+ 15

+ 13

+ 14

+ 14

+ 14

+ 13

Xn .

+ 1

+ 1

- 1

0

+ 1

0

+ 1

- 1

- 1

0

- 1

Temperature of com-

15°

15°

15°

15°

15°

15°

15°

15°

15”

15°

15”

parison

1898.

May.

June.

July.

Sept.

Oct.

December.

4.

5.

9.

28.

13.

17.

23.

10.

14.

14.

16.

X, .

- 4

- 3

- 4

- 3

- 5

- 4

• 6

- 2

- 4

- 8

- 8

Xo .

+ 6

+ 6

+ 6

+ 7

+ 5

+ 5

+ 4

+ 14

+ 10

+ 15

+ 16

Xg ...... .

+ 1

X5 .

-17

- 17

-18

- 18

-20

-20

-21

- 15

-21

- 20

-19

X10 .

1 +15

+ 15

+ 16

+ 15

+ 17

+ 18

+ 18

+ 21

+ 16

+ 15

+ 15

Xn .

- 1

- 1

- 1

- 2

0

0

+ 1

+ 4

- 2

-, 5

- 3

Temperature of com-

15°

15°

15°

15°

18°

18°

0

O

<M

19°

14°

15°

15°

parison

1898.

1899.

1900.

December.

Jan.

March.

1 May.

June.

Oct.

February.

March.

19.

27.

- 17.

24.

6.

16.

25.

. 5.

12.

14.

Xi .

- 8

- 8

- 9

-12

- 9

- 10

-10

- 17

- 14

- 15

Xo .

+ 17

+ 15

! +14

+ 11

+ 15

! +15

+ 17

+ 17

+ 20

+ 14

X3 .

+ 3

+ 1

+ 2

+ 16

—

—

+ 18

—

+ 17

x3 .

-20

- 19

-20

-23

-17

- 18

-21

-23

- 18

-24

X10 .

+ 15

+ 15

+ 14

+ 12

+ 16

+ 18

+ 15

+ 11

+ 16

+ 12

Xu .

- 5

- 3

- 3

- 6

_ g

- 2

- 2

- 8

- 3

- 7

Temperature of comparison .

15°

15°

15°

15°

15°

16”

16”

15°

16°

15°

Y 2

164

DR, H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

original cells from the date of the last comparison given in Table VII., Section 25,
p. 151 ( loc . cit.), to the close of the present series of experiments. Various other
tables are given here in order for the other types of cells, and I have designated
the different cells by capital letters indicating the type, and by a suffix to indicate
the number of the particular cell. The crystal cells are given as X, the sealed cells
by S, the inverted cells by XR, and the cadmium cells by Cd. In the original table
of comparison already referred to, cells Xl5 X2, X3, X5, X0, X10, and Xn were given. Of
these cells, all are at present in existence, with the exception of X6, which was taken
away from the laboratory and since broken. Table I. may be taken as a continuation
of the older table. The relative differences in these cells, although somewhat larger
than is usually obtained in constructing a number of cells from the same lot of
materials, have been maintained so consistently that over extended periods a constancy
of 1 in 100,000 can be easily assumed. Later results show that cells X2 and X5 have
lowered somewhat, but even in these two cases the drop is less than T mv,, and
takes place so gradually as to be easily corrected for. It is highly probable that all
these cells will eventually become lower in value as they become older, on account of
the drying up of the crystals.

During the winters of 1897-98 and 1898-99 a number of tests were made by some
of the advanced students on cells prepared by themselves under my supervision.
These cells were all subjected to a temperature cycle of 15° to 0°, to 15° to 30°, to 15°.
The first batch of cells made during 1897-98 were in the inverted form, and were
made in the usual way in a long test-tube for immersion in the water-bath.
A 10 per cent, zinc amalgam was placed in the bottom of the test-tube and covered
to the depth of about 2 centims with moist zinc sulphate crystals. The paste of
mercurous sulphate and zinc sulphate crystals placed on top of the crystals was made
in the usual way by mixing moist crystals with pure washed Hg2S04. The positive
electrode consisted of a platinum wire flattened at one end, amalgamated, and
inserted in the paste. The wire was protected by a small glass tube and reached to
the top of the test-tube, where the glass was melted around the wire to form a
mercury cup. The negative electrode was a platinum wire protected in a similar way
and thrust into the amalgam while still warm before the cell was filled with the other
ingredients. The cell was sealed by shoving a cork down the test-tube, with the
two electrodes passing through holes made for them, to within a few millims. of
the ingredients. The cork was about 1 centim. thick, and was sealed by inserting
particles of marine glue and melting them in place by carefully warming the glass
over the cork. The crystals of zinc sulphate were prepared by re-crystallizing the
ordinary pure heptahydrate salt after neutralization with zinc oxide, and treating
with a small quantity of the washed Hg„S04 when in solution in the usual way.

The cells made in 1898-99 were of the older type, with an amalgamated zinc rod
with positive electrode at the bottom of the test-tube. They differed from the
original crystal cells in having an amalgamated flattened platinum wire in place of

BETWEEN THE FREEZING AND BOILING-POINTS.

165

the metallic mercury. In preparing the crystals for these cells it was deemed
unnecessary to follow the old prescription, inasmuch as they were made from the
purest anhydrous salt purchased from Merck. However, a few of the cells were
made from crystals that had been treated when in solution with a small quantity
of Hg2SOj,, but filtered out before re-crystallizing.

In the table of comparison now given the inverted cells are expressed as difference
from the mean of the five crystal cells Xl3 X2, X3, X10, and Xn. The temperature
changes between 15° and 0°, and 15° and 30° are also given, as determined by the
different students who made the cells.

Table II. — Comparison of Inverted Cells to Mean “ Crystal ” (1897-98).

Cell.

Difference from mean
crystal in mvolts.

E.M.F. changes between

15° — 0°.

15°— 30’.

XRn .

+ 0-16

+ 16-67

- 19-45 -

XRx, .

+ 0-17

16-67

19-46

XR,3 .

+ 0-20

16-69

19-58

xr]6 .

+ 0-19

16-66

19-60

XRir .

+ 0-10

16-67

19-61

XR1S .

+ 0-20

16-69

19-55

XRin .

+ 0-24

16-60

19-51

Means

0-166

16-67

19-54

These tests are sufficient to show that the inverted cell gives a value somewhat
in excess of the older crystal cells. The temperature change between 15° and 0°
is also somewhat larger than the value given by the crystal cells, which was
+ 16 ‘62 mvolts. The reason for this may possibly be, as I have already pointed out
in another place, that the sensitive electrode (negative) is at the bottom of the cell
and deeply immersed in the bath, whereas in the crystal cells the negative electrode
is a zinc rod at the top of the cell, and although immersed below the level of the
liquid in the temperature bath, may yet conduct an appreciable amount of heat and
be at a slightly different temperature to that of the cell.

I have used the term sensitive for the negative electrode because a small difference
of temperature between it and the other parts of the cell influences the E.M.F. very
considerably. This can be very forcibly shown in the case of an inverted cell by
removing it from the 15° bath and standing it on a cold surface. This produces
a larger change (increase) in the E.M.F. than if the complete cell were immersed at
the cooler temperature ; and in a similar way for a higher temperature, the negative
electrode being warmer than the other parts of the cell, the E.M.F. decreases more
than it would were all the cell at the same temperature. Hence, in the crystal cell,

166

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

when immersed at a temperature above or below 15°, the possibility of the zinc rod
being at a slightly different temperature to that of the cell is thinkable, and would
act in such a way as to make the positive change between 15° and 0°, as well as the
negative change between 15° and 30°, appear smaller than the true values.

Later on in the year, on September 10, a comparison was made of the old crystal
cells with two of the original sealed cells, made in 1896, and at that time still in the
possession of the laboratory, as well as three new sealed cells. This comparison is
given in Table III. and gives the differences in millivolts between each sealed cell and
the mean of the old crystal cells, the comparisons of which were given in Table I.

Table III. — Comparison of Sealed Cells with Old “ Crystal ” Cells.

Cell.

Difference from mean of old cells.

Si .

S5 .

Sn .

nSn .

S12. . . . .

+ 0 • 18 mv. )

+ 0'34 ” ' Mean

+ 0-20 „ y 90

+ 0-23 „ | +°’23mv.

+ 0'21 „ J

The agreement of the new sealed cells with the old sealed cells is good, but both
show that the mean of the old crystal cells is too low. This makes the mean of the
sealed cells, on comparing Tables II. and III., agree with the mean of the inverted
cells to '06 mv.

A comparison of the six crystal cells X1} X2, X3, X5, X10, and Xn, with six of the
newer crystal cells made by the students in 1899, was made on March 14, 1900,
when the last comparison of the old crystal cells given in Table I. was obtained. In
Table IV. this comparison of the six new cells is given, and the differences expressed
in millivolts from the mean. Cell X2 is included, and differenced from, but not
included, in the mean.

Table IV. — -Comparison of New “ Crystal ” Cells, on March 14, 1900.

Cell.

Difference from mean.

x21 ....

- 0 • 24 mv.

x23 ....

+ 0-05 „

X27 ....

-0-05 „

X99 . . . .

+ 0-05 „

X31 . . . .

+ 0-11 „

X33 . . . .

+ 0 ■ 08 ,,

x2 . . . .

-0-17 „

During the winter previous, when all these new cells were made, the tests on
the temperature changes between 15° and 0°, and 15° and 30° gave the mean

BETWEEN THE FREEZING AND BOILING-POINTS.

167

values -f- 16 '60 and — 19 '45 respectively, which agree very closely with the values
given by the old cells, but are smaller than the values given by the inverted cells.
X2 in the above table is seen to be lower than the mean of the new crystal cells
by — '17. Under date March 14, in Table I., it is seen to be + T4 mv, above the
mean of the old crystal. This would make the mean old crystal lower than mean
new crystal by '31 mv. This excessively low value is influenced by ceils X5 and X2
on the mean, which have apparently gone down since the earlier tests. If we reject
these two cells from the mean, as being too low, the mean value of the four remaining
cells is increased by TO mv. and the difference between mean old crystal and new
crystal reduced to '21 mv. In Table II. it was seen that the inverted cells in
January, 1898, were '17 mv. higher than the mean old crystal. If we reject
cell X- from the mean of the old crystal cells, as being; too low, then the difference
between the two sets of cells is reduced to T2 mv. Also in Table III. the mean of
the old cells is seen to be '23 mv. lower than the mean of the new and old sealed
cells. If we reject X5 as before from the mean,, the difference is reduced to T 7 mv.

If now we can assume that the mean inverted and mean sealed was the same as
the mean new crystal (which unfortunately could not be verified by a direct
comparison), then we see that the old cells have lowered in value since January,
1898, by '06 mv., or 4 parts in 100,000. Another indication that the cells have all
lowered somewhat in value is afforded by a comparison as early as 1896 with six
sealed cells, including SL and S5 of Table III. The mean value of the sealed cells
was '08 mv. higher than the mean old crystal including cell X5, which was more
nearly in agreement with the mean at that time. (See Table VII., page 151, loc. cit.)
It is evident that for some reason the old crystal cells, even from the first, are lower
than what may be taken as the true Clark-cell value, if we may assume that the
mean old sealed, mean new sealed, mean inverted, and mean new crystal are all
within a few hundredths of a millivolt of each other, and of the true Clark-cell
value.

In 1896 the old crystal cells were lower than the sealed cells by '08 mv. ; in 1898
lower than the inverted cells by '12 mv., than the new sealed cells by T7 ; and in
1900 lower than the new crystal cells by '21 mv. This indicates that the mean
value of the old crystal cells is T4 mv. lower than the most probable value that we
can assume, combining all our Clark cells, and this within the limits of error of
perhaps '02 mv.

We can now, from the table of comparisons, assign individual values for the two
Clark cells which were used throughout in the present investigation. These cells
were X2 and X10 of the old crystal cells. From January to December, 1898, X2 was
'08 mv. higher than the mean of the crystal cells, and.X10 was at the same time
'15 mv. higher. If we neglect cell X5 from the mean, as being too low, then this
gives for cells X2 and XL0 in series the values — |- '03, + T0, or + T3 mv. above the
mean. But as mean crystal, neglecting X6, is T4 mv. lower than what we have

168

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

reason to believe is the true Clark- cell value, two mean crystal cells in series would
be '28 mv. lower. Therefore, cells X3 and X10 in series are lower than the true
value by ‘15 mv. During 1899, cell X3 was on an average ‘15 mv. above the mean
crystal cells. Neglecting X5 but not X: this difference is reduced to ‘10. X10 was

also -f‘15 or, neglecting X5, ‘10. This gives for X, + X10 the value +‘20 mv.,
or different from the true Clark-cell value by —‘08 mv. During the early part of
1900, cell Xj commenced to go down more rapidly, partially through the introduction
of cell X3 again into the mean. If we take the comparisons on February 5th and
March 24th, we find for X2 the value +‘16 mv., and for X10 +‘12 mv. Neglecting
X1 as well as X- from the mean, the value of X2 + X10 becomes -f '06 + '02, or
equal to -j- '08 mv., the difference from the true Clark-cell value being now
— '20 mv. Summarizing we have, if e represents the true E.M.F. of the Clark cell
in volts

In 1898 X2 + X]0 2 Xe— '00015 ]

In 1899 ,, —’00008 Mean value —'00014.

In 1900 „ „ - '00020 J

This gives the mean error, if we assume from 1898 to 1900 the mean value
2 X e — '00014 as representing the true E.M.F. of the two Clark cells used in these
measurements, as in

1898, + 4 X 10-6 ; 1899, — 2 X 10“5; 1900, + 2 X 10“5 ;

all of which are less than 1 part in 10,000, and outside the possible limits of error
of the other measurements.

Whenever the temperature of the Clark cell was other than 15°C.,the E.M.F.
was calculated, assuming a value at 15°, by the temperature formula obtained
between 0° and 30° for a mean temperature of 15° which has already been given.
As a matter of verification of this formula, which was deduced from the old crystal
cells, I have summarized in Table V. the observations that have been made since,
both by myself and the students under my supervision.

Table V.

Type of cell.

Change in millivolts between —

15° and 0°.

15° and 30°.

Old B.O.T. crystal .

1896

+ 16-62

- 19-48

Sealed .

1896

+ 16-62

-19-58

Portable B.O.T. crystal .

1897

+ 16-60

-19-40

Inverted .

1898

+ 16-67

-19-54

Portable B.O.T. crystal .

1899

+ 16-60

-19-45

Temperature formula .

—

+ 16-60

-19-40

Values obtained by Dr. Kahle

—

+ 16-40

- 19-40

BETWEEN THE FREEZING AND BOILING-POINTS.

169

The values obtained by Dr. Kahle at the Reichsanstalt are somewhat lower.
They were obtained for the H-form of cell with negative electrode, zinc amalgam.

The portable crystal cell in Table V. refers to the case where the metallic mercury
for positive electrode is replaced by a flattened platinum wire amalgamated. The
portable cells in Table VII., original paper, p. 152, made by the students in 1897, and
from which the tests (in 1897) in Table Y. are taken, were compared with the old
crystal cells rather too soon after setting up, to use in determining the true value of
the E.M.F. of the old cells, as we have done for the latter cells and the sealed cells
in 1896. The mean value of all these cells is very close to the crystal cells, but later
tests showed that some of them gave too low a value at first.

Ratio of Clark to Cadmium Cells. — In 1897, to check the value of the Clark cells
made by us in the laboratory, several Weston cadmium cells were constructed.
These were made in the inverted form, and one was made in the H-form after
type III. described by Jaeger and Wachsmuth (‘ Wied. Ann.,’ vol. 59, p. 580, 1896)
in their paper on the cadmium cell. All the cells had a cadmium amalgam of 1 to 6
proportion, as recommended by Jaeger and Wachsmuth, except two, which were
made after the B.O.T. “crystal” cell type with cadmium stick. These two cells,
however, as was expected, gave much too high an E.M.F. and were only made as a
matter of interest. I have described these cells in another place (c Journ. Phys.
Chem.,’ vol. 4, 1900), with comparisons which were obtained in 1897.

The temperature coefficient obtained for these cells was a little in excess of that
found by Jaeger and Wachsmuth for their cells, but is more in agreement with the
value found by Dearlove (‘Electrician,’ vol. 31, p. 645, 1893) and the original
value given by Weston. The expression is a linear one, and reads

E, = E15 - -086 ( t - 15°),

and holds with great accuracy over the range 15° to 40° C. At 15° a change of
state occurs in the cadmium sulphate, so that no formula can be made to hold below
that point.

I made a determination of the ratio of these cadmium cells to the old crystal cells,
by means of the cylinder potentiometer and 6000-ohm galvanometer which were used
in the earlier comparisons of the Clark cell, given by Professor Callendar and
myself in our original paper (p. 121). The potentiometer was repeatedly calibrated
by comparison with the Thomson-Varley slide potentiometer, described in another
place. The corrections for uniformity were somewhat large, but were exceed¬
ingly consistent, and were determined by myself, as well as by a large number
of the students in the ordinary course of their work. The cells, both cadmium and
Clark, were immersed at a constant temperature near 15° throughout the test.
Table YI. contains the result of this test.

* Callendar, ‘ Phil. Trans.,’ A, 1902, p. 63.

VOL. CXCIX. — A, Z

170

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

Table VI. — Comparison of Clark and Cadmium. March 6. 1897.

Cell.

Potentiometer reading
corrected to 15°.

Corrected for
uniformity.

X] ....

68225

68290

Cd3 ....

48453

48575

Cd4 ....

48460

48582

Cd, ....

48458

48580

Cd„ . . . .

48460

48582

Correcting reading of X} to mean of old crystal cells and reducing mean cadmium
reading to 20° by the formula,

E, = E15 - -086 (t - 15°),

the ratio of Clark to cadmium becomes

Clark 15° _ 68294

cadmium 20° 48558

1-40644.

The ratio obtained by Kahle for the cells in the possession of the Reichsanstalt
was (‘ Wied. Ann.,’ vol. 67, p. 35, 1899),

Clark 15°
cadmium 20°

1-40663.

The value of our ratio is somewhat lower than the value given by Kahle, which
may be explained by either assuming the cadmium cells too high or the Clark cells
too low. We have seen, however, that the mean of the old crystal cells is lower
than the most probably true Clark-cell value obtained by comparison with later tests
by "14 mvt., or 1 part in 10,000.

Correcting the ratio by this amount, it becomes 1*40658, a value nearly identical
with the value obtained by Kahle.

The Absolute Value of e. — The assignment of the true value of e to the cells used
in the present work is, at present, somewhat difficult. Glazebrook and Skixner
found on standardizing the B.O.T. form of test-tube cell by means of the silver
voltameter, and assuming the value "001118 gram. -sec. for the electro-chemical
equivalent of silver as determined by Lord Rayleigh and Mrs. Sedgwick, that the
value was 1"4342 international volt at 15° C. More recently we have the
measurements made by Dr. Kahle at the Reichsanstalt with the Helmholtz Electro-
dynamometer (‘Wied. Ann.,’ vol. 59, p. 532, 1896, and ‘ Zeit. fiirlnstk.,’ June, 1898),
which give a result independent of the value assigned to the silver voltameter. We
have also the value obtained recently by Professors Carhart and Guthe, at Ann

BETWEEN THE FREEZING AND BOILING-POINTS.

171

Arbor, Michigan University (‘ Physical Review,’ vol. 9, p. 288. 1899), with a type of
dynamometer designed by themselves. The results of these measurements show a
wide divergence. The values found are

Glazebrook and Skinner . . . 1-4342 volt at 15°.

Kahle . . . 1-43285 ,, ,, ,,

Carhart and Guthe . 1-4333 ,, ,, ,,

The large discrepancy in the value of the mechanical equivalent of heat obtained
by the electrical methods used by Professor Griffiths and Professors Schuster and
Gannon, as compared to the value given by the direct mechanical method, has so far
hinged on the value to be assigned to the Clark cell. The older, and for so long a
time accepted, value, 1*4342, there is every reason now to think is too high. The
value given by Dr. Kahle, i.e., 1 '43285 volt, is at the same time probably a little too
low. The value found by Carhart and Guthe depends on the mean of three
determinations differing in the extreme by "5 mvt. These three determinations were
made for two Clark cells in series, one of which was afterwards compared to the
Reichsanstalt cells and found to be in good agreement.

At present there is a grave uncertainty in the absolute value of this fundamental
constant, which requires immediate attention. It has been pointed out that the
value of the mechanical equivalent of heat found by Griffiths would be brought
into harmony with the values found by Rowland by the direct mechanical method,
by assuming the Clark cell 2 mv. lower than the value found by Glazebrook and
Skinner. The value found by Schuster and Gannon requires a somewhat smaller
correction in the same direction.

In the face of these uncertainties in the value to be assigned to e, I have adopted
the older value, 1*4342 international volt, as the basis of my calculations of the
absolute value of the mechanical equivalent, in order to bring my results into
comparison with those of Griffiths and of Schuster and Gannon. On this basis
I have had the temerity to combine the mean value of the mechanical equivalent
obtained by integrating the curve of absolute values between 0° and 100° with the
determination of the mean value obtained by Reynolds and Moorby, and have
obtained by that way an absolute measure of the Clark cell in terms of the
mechanical units, which is probably as accurate a value as has yet been obtained,
provided the values assumed for my resistance standards are correct.

The discrepancy in the two values of the mean mechanical equivalent, the one
obtained by integrating the variation curve, and the other obtained as a direct
determination, is "132 per cent. As I have used the value e of the Clark cell in my
measurements squared, this reduces to "066 per cent, on 1 "43420, and shows that
the value assumed for my cells is too high by this amount. The true value of the
Clark cell I have assumed for calculation is 1"4342, which would give for the two

z 2

172

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

cells X3 -f- X10 in series, the value 2 '8 6 8 4, pitmded they were equal to the true
value. We have seen, however, that these two cells differ from the most probable
true value of the Clark cells made in this laboratory by 00014 volt, which would
give for the true value of X3 + X10 2-86826., or 1-43413 each. Reducing this value
by ’066 per cent., we have, as the value of each of my Clark cells,

1 "4331 8 iut. volt,

and the most probable true Clark-cell value

1-43325 int. volt at 15° C.

which is in remarkable agreement with the absolute measurements of Carhart and
Guthe for their Clark cells.

From the ratio of the Clark to cadmium, the value of the cadmium cell is found
to be

1-01895 int. volt at 20 C.

Sec. 3b. — Measurement of Resistance.

Next in importance to the value of e for the Clark cells, which we assume for
the calculation of the absolute value of the mechanical equivalent of heat, is the
value to be assigned to R for the resistance used in these experiments.

At the outset we were exceedingly fortunate in having the laboratory equipped
with a large number of 1-ohm resistance standards certified by the Electrical
Standards Committee of the British Association, which were obtained in 1893. The
work, therefore, of standardizing the resistances which were made for the present
series of experiments was reduced to a minimum by the facility with which they
could be compared to these standards on a Nalder type of Carey-Foster commutator-
bridge. This bridge was supplied with a set of ratio-coils and bridge-wires which
could be interchanged at will, and selected to be comparable in size with the
resistances compared. During the first experiments which we made on the specific
heat of mercury and the early trial experiments with the water calorimeter, the
electric heating current was passed through a 1-ohm manganin coil for standard,
which was immersed in paraffin oil. The difference of potential across the terminals
of this specially- constructed resistance was of the same order as that across the column
of liquid in the fine flow-tube in the mercury-calorimeter, and also equal to that across
the platinum heating-wire in the first water-calorimeter. This was arranged for
convenience in balancing on the potentiometer.

The coil was made from two manganin wires, 1 millim. in diameter, connected in
parallel and wound on an ebonite frame. Connections were made to the coil at the
bottom of the frame, which was held vertical, by two heavy copper-wires, ^ inch in
diameter, so arranged as to have 3 or 4 inches immersed in the oil-bath with the coil.

173

BETWEEN THE FREEZING AND BOILING-POINTS.

The wires were bent into an inverted U and made to fit into mercury- cups, either on
the commutator-bridge or in the main calorimeter circuit from which the potential
terminals were taken to the potentiometer. For the later experiments with the
water calorimeter it was found advisable to alter the resistance of the heating- wire to
-5 ohm, so that another mangaiiin-resistance was made similar to the first one and
connected in parallel with it in the calorimeter circuit. Our numerous comparisons of
these ohms with the certified standards were far from satisfactory, but the cause was
at first sight not apparent.

The resistance of both the coils was found to increase, after carrying currents of
from 4 to 8 amperes in a series of experiments, of the order of 2 or 3 parts in 10,000
in two weeks. This was somewhat annoying, and necessitated repeated comparisons
with the standards and numerous corrections. It was also a matter of doubt whether
the resistance of the coils remained the same when the heavy currents were passing,
seeing that they produced such a large permanent change in the resistance. We
finally commenced to suspect the real cause of the trouble to be at the point where
the manganin-wire was soldered on to the heavy copper-wire. In the face of this
uncertainty it was decided to abandon these resistances altogether for others made of
platinum-silver wire according to a different design. Both these new 1-ohm resistances
have proved to be so reliable and constant since they were made, in May, 1898, that
it is proposed to give a short description of them here. They were both made on an
exactly similar design.

The frame-work consisted of two heavy plates of mica, 4" X 2 W, placed side by
side, and separated about § of an inch by ebonite strips at each narrow end. Both
ebonite strips were split from end to end, parallel to the
mica plates and half-way between them. The strips
were fastened to the mica plates by ebonite washers and
small screws, shown in fig. 4, which gives a general
view of a resistance. The plates were arranged so that
they could be separated or put together quickly by
removing two screws at either end, clamping the ebonite
strips together. Two jjtench copper-wires were passed
through holes bored for them through the splits in the
ebonite strips at each end, in such a way that they
were clamped in place by the ebonite. These heavy
wires, when in place, connected the space between the
mica plates with the outside of the frame-work. At
each end the two wires were bent at right angles so as
nearly to meet, and were inserted and soldered into
holes made for them on the opposite faces of a small
copper block. Heavy copper-wires (^-inch) were soldered into holes in these copper
blocks and bent into an inverted U for connecting to the commutator-bridge.

174

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

Each of the two mica plates was serrated on the two long edges and two bare
platinum-silver wires, '4 millim. in diameter, wound on side by side. After winding,
the ends of the platinum silver wire were fused to copper- wires of the same size in a
blow-pipe flame. The wires on the frames were then annealed at a low red heat by
passing a heavy electric current through them. After the two mica plates were
clamped together so as to include the heavy copper- wires at both ends, the copper-
wires that were fused to the platinum-silver wires were soldered to the end faces of
the copper-wires protruding into the space between the mica plates. There being in
all eight ends to be soldered and four heavy copper-wires to solder into, each large
copper-wire was connected to two of the small copper- wires fused to the platinum-
silver wire. Each 1-ohm consisted thus of four bare platinum-silver wires, 16 millims.

Fig. 5.

in diameter, in parallel ; direct solder joins of platinum-silver with copper were
avoided, and the mica plates were arranged so as to give the best possible circulation
when immersed in an oil-bath.

The paraffin oil-bath was made from a square ebonite box, and included, besides the
two 1-ohm coils, a stirrer and coil of metal tube for a water circulation, fig. 5. The
coils always remained fastened in the bath, and when it was necessary the bath,
including the ohms, could be removed from the position assigned for it in the experi¬
ment where the ends of the two inverted U-shaped connections from the coils were
immersed in mercury-cups in two heavy copper forgings in the calorimeter circuit.
When a comparison was made on the commutator-bridge, the bath was conveyed to
the place where the bridge was always kept. During a comparison, the stirrer was
run by a small electric motor and the temperature of the oil taken by a thermometer
immersed in the bath. For the determination of the temperature coefficient of the

BETWEEN THE FREEZING AND BOILING-POINTS.

175

coils, water at different temperatures was run through the circulating tube so as to
change the temperature of the oil.

Up to the time of writing, I have been unable to compare these ohms directly with
official standards. It will, therefore, be necessary for me to describe a series of
comparisons of these ohms with eleven certified ohms in the possession of this
laboratory, together with a standard 1-ohm coil from the German Beichsanstalt, sent
us for comparison by the Physical Department of the Massachusetts Institute of
Technology. By means of these comparisons, we may possibly arrive at a result for
the value of B somewhere near the truth.

Throughout the present work I have used only one of the certified standards
(No. 4086) to check the constancy of the two platinum-silver ohms, as it was of a
better and more convenient form to use on the bridge than the others and had a
much smaller temperature coefficient.

Before describing the tests I will briefly review the method of comparing the ohms
and the method of finding the value of the bridge-wire used on the Carey-Foster
bridge throughout these tests. The 1-ohm pair of coils supplied by the makers of
the bridge were used for the ratio coils, and a bridge- wire having a resistance of
about ‘002 ohm per centim. was used. The bridge-wire was just 10 centims. long,
with a scale graduated into half centimetres and millimetres. A lens was also
supplied for reading the position of the balance point. The galvanometer for
obtaining the balance point was a very sensitive 9-ohm Thomson reflecting galvano¬
meter, which was used for the thermometer work. It had a telescope and scale, and
was sensitive to 50 scale-divisions for 1 millim. of bridge scale, which, of course, was
far more sensitive than was required, or even quite convenient to work with. The
current supplied to the coils was from one accumulator through 20 ohms external
circuit. By simply lifting the commutator from the mercury cups on the bridge and
revolving it through half a revolution, the connections could be made so as to reverse
the position of the two resistances relative to the ratio coils. If P and Q are the
ratio coils, B and S the resistances to be compared, then, when the current is reversed
in B and S, but not in P and Q, we have, B/S — S/B = p(dl — d.2), independent of
P and Q.

Here dl and d2 are the readings of the balance points on the bridge-wire, and p a
constant to reduce to ohms.

Let B/S = (1 -j- r), where r is a small quantity ; then S/B = (1 — r) and

B/S — S/B = 2 r — p [dY — d.2).

To find p, S may be changed to S} by shunting with a known large resistance, say
100 ohms.

We have then, if B/Sj = 1 + r + dr, Sx/B = 1 — r — dr, and reversing the
current as before,

176

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

from which

therefore

R/S2 — Sj/R = 2 (r -f dr) = p (d?j — dx),

2clr

(d, — dx) - {dx — d.2) ’

dr (dx -d,)

{ds - dp - (dx - d,) ’

the value of dr being ( S — ^ )•

& \ S + 100/

This gives for R the value S(1 + r), where S is the known standard. The value
of p was obtained a number of times, both by myself and a number of the students.
The values obtained since 1897 are in ohms — '001022, '001015, '001019, '001011,
•001028, '001028.

This gives a mean value of '001020 ohm per division for bridge-wire C.

The following is a list of the standard 1-ohm coils used in the comparisons. Each
coil had a certificate signed by the secretary of the Electrical Standards Committee,
and dated either in 1892 or 1893.

List of Certified Standards.

Platinum-silver Wire Coils (embedded in Paraffin Wax).

O

No. 3565 certificate, '99957 true ohm at 16 '4.

„ 3566

„ '99960

55

16-5.

„ 3567

„ '99949

55

16-4.

„ 3568

„ '99961

55

16-5.

„ 3569

„ '99964

55

16-5.

„ 3402

„ '99971

55

16-7.

„ 3403

„ '99967

55

16-5.

„ 3404

„ '99970

55

16-7.

„ 3405

„ -99960

55

16-3.

„ 3406

„ -99960

5 5

16-3.

Manganin Wire Coil (in Oil-bath).

O

No. 4086 certificate, '99978 true ohm at 15'9.

Reichsanstalt Standard.

No, 1214 marked 1 true ohm at 20° C,

BETWEEN THE FREEZING AND BOILING-POINTS.

177

From 3565 to 3406 each ohm was of the older form, with the wire embedded in
paraffin wax and made to insert in a water-bath, with long heavy wire connectors for
the terminals of the Carey-Foster bridge. No. 4086 was the best form to use with
the bridge, as the method of having the other coils always in paraffin wax is bad, and
it is never jmssihle to know exactly the true temperature of the coils. The German
standard was evidently made of manganin wire on account of its very small
temperature coefficient. Unfortunately coil 4086, to which all of my results were
referred (‘ B.A. Report,’ 1899), seems to be different to the others by as much as
6 parts in 10,000. It is difficult to see how it could have been injured in any way
since it came into the possession of this laboratory, and, as will be seen presently, the
comparisons of this ohm with both of the specially constructed platinum-silver ohms
does not indicate any possible change since May, 1898. The cause that has been at
work to alter its resistance has left it entirely unaffected during the last two years.

Two tables of comparisons are now given of all the 1-ohm coils. The first set in
Table VII. was taken by myself and expresses all the ohms, except 3566 and 1214,
in terms of 4086. For the second set in Table VIII. I am indebted to Mr. Fraser.
For reducing the values of the ohms to one temperature, temperature coefficients
were used which were obtained either by myself or Mr. Fraser in duplicate by
special experiment, and verified repeatedly by the students. All the platinum-silver
standard ohms were found to have the coefficient + '000254 t°.

The manganin ohms 40S6 and 1214 were found to have the coefficients '000018
and -f- '00M022 respectively.

The different columns of Table VII. are arranged so that the first gives the
number of the ohm, the second the certified resistance at temperature given in the
third column, and the fourth column gives the length of bridge-wire multiplied
by '001020 to reduce to ohms, which represents the difference in resistance between
each ohm and No. 4086. In the fifth column is given the temperature of the
different ohms during the comparison, and in the sixth the temperature of 4086.
The seventh column contains the value of 4086 at the temperature of comparison
found from the certified value by the temperature coefficient. As all the platinum-
silver standard ohms were larger than 4086, the eighth column is obtained by adding
columns 4 and 7. This gives the resistance of each ohm in terms of 4086. In the
last column, for comparison, I give the value of each ohm in terms of its own
certificate, and corrected to the temperature of comparison in column 5 by the
temperature coefficient. The values in the eighth column in terms of 4086 are all
systematically smaller than the values in the ninth column, whereas they should be
equal. The observations differ amongst themselves somewhat, but they are as good
as can be expected from the difficulty of knowing the true temperature of the coils
embedded in the wax.

In all these tests the standard ohms were left for several hours near the place
of test, so that they could assume, as nearly as possible, the temperature of the air.

VOL. CXCIX. - A. 2 A

Table YII. — Comparison of 1-ohm Standards in Terms of 4086.

178 DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

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BETWEEN THE FREEZING AND BOILING-POINTS.

179

A thermometer placed in the hole extending through the middle of the embedded
coil was taken as the temperature of the coil. As No. 4086 was arranged with
a stirrer, a thermometer could he placed in the oil in contact with the coil, and the
true temperature obtained. The current used on the bridge was not sufficient to
cause j)erceptible heating.

Table Till, contains the comparisons of the ohms, at an entirely different tempera¬
ture. These tests were made in the basement of the building, where the temperature
was considerably lower than where the tests in Table All. were made. In this case
also the coils remained at least a day or two at the temperature of test, and did not
vary to any extent from that. The table is arranged as in Table VII., only the com¬
parisons were made in terms of No. 3569. This shows a very good agreement of all
the platinum-silver standard ohms, including the Reichsanstalt ohm, 1214, but shows
that by assuming the corrections of 3569, the value of 4086 is very much above that
given in its certificate. This difference indicates an error of ‘00059 ohm assuming
3569 as correct, or ‘00065 referred to the mean of all the ohms. In Table VII.
we saw that the platinum- silver ohms were all lower than their certified values
when calculated assuming 4086 to be correct, the mean difference being ‘00052.
These differences are both in the same direction as regards the relationship of 4086
to the other ohms. The difference of 1 in 10,000, obtained by Mr. Fraser and
myself between the two values, i.e ., ‘00065 and '00052, must be ascribed to the wide
difference in temperature of our respective tests, as well as to the uncertainty ot
knowing accurately the true temperature of the paraffin-embedded coils.

We are forced now either to accept the certificate of 4086, and reject all the other
11 ohms as being in error, including the Reichsanstalt Standard, or to reject the
certificate of 4086, and accept the certificates of all the others. The alternative of
giving 4086 equal weight in the mean seems to be hardly justifiable considering the
mass of evidence against it.

I have decided to reject the certificate of 4086, and I have accordingly corrected
it in the following way : in terms of the platinum-silver standard ohms, 4086 is
equal to its certified value +‘00052 by the comparison made at 22° C. By the
comparison made at 13° C. it becomes equal to its certificate +‘00065. By
comparing directly with 1214, the value of 4086 becomes equal to its certificate
+ ‘00056 in one test, and +‘00061 in another test, or equal to +‘000585 in the
mean. This agrees very closely with the mean value of the two separate determina¬
tions with the other ohms, which comes out +'000585. We may, I think, then
safely assume that the value of 4086 is equal to its certified value +‘00058, which
comes out ‘99978 + ‘00058 at 15°‘9, or 1‘00036 + ‘000018 (20° - 15°‘9), or equal to
1 ‘00043 true ohms at 20° C.

A summary of the various comparisons made of the two new platinum-silver ohms
is given in Table IX. in terms of 4086, assuming for convenience that it is exactly
1 ohm at 20° C. The resistance of each ohm is reduced to 20° C. in column 4 of each

o A o
u A u

180

DE. H. T. BAENES ON THE CAPACITY FOE HEAT OF WATER-

set by means of the temperature coefficient found from the tests given in Table X.
Most of the comparisons up to September 10, 1898, were made by Mr. Sheffield,
and from that date on, by myself. The maximum variation from the mean is
5 X 1 0 ~ 5 ohm, and is within the limits of error for a series of comparisons
such as these. As a rule the agreement is very much closer than this. Taking
the value of 4086 as equal to 1 ’00045 true ohms at 20° O. in place of the value
assumed for calculation in the table, we find Coil 1, 1 ‘00132 -f- ‘00043 = 1 00175
true ohms at 20° C., and Coil 2, 1 ‘00043 ohms. At any other temperature the
coefficients +‘000250 for Coil 1, and ‘000246 for Coil 2, are used, which were
obtained from the experiments detailed in Table X.

The value of the two 1-ohms in parallel is very easily determined on a small slide-
rule, by assuming the ohms equal to (1 + dx) and (1 + cl.2) respectively, where +
and + are equal to the small differences from unity, then

(1 + dj) (1 4 A) _ _ _ jr (A + A)

(1 + A) + (1 + A) 2 + + + +

neglecting products and powers of dx and oh.

At 20° C. the value of the fraction is

R = ‘5 + = '500544 ohm.

At 10° C., when No. 1 is equal to (1 — ‘000 77), and No. 2 (1 — ‘00205),

R = ’5 - = '499294 ohm-

We may accept then for calculation the most probable value of the two platinum-
silver ohms in parallel to be

‘500544 true ohm at 20° C., ‘499294 true ohm at 10° C.,

where one true ohm = 1 ‘01358 B.A, unit, as given in all the certificates of the
standard ohms.

Current Heating. — It is a matter of importance to determine the true resistance
of the two coils when the maximum current used in these experiments was passed
through. For the largest flows of water, when the largest heating current was
required, this amounted to 8 amperes. This current was divided between eight
•4-mill im. platinum-silver wires immersed bare for their entire length, about 1 metre,
in oil, which was vigorously stirred. Each wire was required to carry then only
1 ampere, or develop only 4 watt-seconds heat energy. It was impossible to imagine
that the temperature of the wire could have been sufficiently different to that of the
oil to appreciably affect the resistance. A difference of ‘1° between the wire and oil

181

BETWEEN THE FREEZING AND BOILING-POINTS.

Table IX. — Comparison of Platinum-Silver ohms, with 4086 taken as

1 ohm at 20° C.

Coil 1.

Coil 2.

Date of
comparison.

Tempera¬
ture of
comparison.

Resistance
at tempera¬
ture of
comparison.

Resistance
corrected to
20° C.

Tempera¬
ture of
comparison.

Resistance
at tempera¬
ture of
comparison.

Resistance
corrected to

20° C.

1898.

0

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May 26th .

19 • 2

1-00113

1-00133

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1-00131

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1 -00128

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15-6

1-00022

1-00132

15-9

•99900

1-00000

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1-00071

1-00133

17-6

•99939

•99998

PP

August 9th .

19-4

1-00118

1-00133

19-5

•99988

1-00000

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17-0

1-00059

1-00134

17-0

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1-00003

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October 4th .

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1-00135

15-3

•99889

1-00004

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1-00056

1-00133

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18-27

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1 00132

18-46

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1-00001

Table X. — Temperature Coefficient of Platinum-Silver ohms.

Coil 1.

Coil 2.

Temperature
of Pt-Ag
coil.

Difference from
4086 at 20° in
ohms.

Calculated
from curve.

Temperature
of Pt-Ag
coil.

Difference from
4086 at 20° in
ohms.

Calculated from
curve.

°

211

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1 82

DR. H. T, BARNES ON THE CAPACITY FOR HEAT OF WATER

would have produced an error less than '00002 ohm, or less than 2 parts in 50,000,
whereas it is probable the actual difference in temperature did not make an error
one-tenth of this amount.

Sec. 3 c. — Measurement of Temperature.

By far the most important factor that determines the character of the curve for
the variation of the specific heat of water with temperature is the particular
thermometric scale to which the results are referred. This was most forcibly brought
out by Rowland in his memoir, and it was pointed out by him that without the
greatest care in reducing his mercurial thermometers to the air scale, the value of the
specific heat of water would have apparently remained constant in terms of the
mercurial scale over the range of his experiments. The discovery of the rapid
decrease in specific heat with increase of temperature from 0° to 30° C. was only
made through this careful reduction.

In the present series of experiments there were no thermometric difficulties such
as are to be met with in the use of a mercurial standard owing to the use of platinum
thermometers. In working to the 1 0,000th part of a degree Centigrade, such
corrections as a change of zero, pressure on the bulb, capillary and stem corrections,
are so large in the case of the mercurial standard, that for large intervals of
temperature the readings are far from reliable. With the platinum thermometer we
still have to deal with the question of a change of zero and a stem correction, but
these are so small that with sufficient care they may be eliminated altogether.

In speaking of these possible sources of error in connection with the measurement
of temperature with the platinum thermometer, I am referring to a limit of accuracy
seldom required in most determinations. The first source of error is already well
known, and has often been subject of controversy over the reliability of the platinum
thermometer, though chiefly, I am convinced, by those who are either prejudiced or
who require more experience in this class of work. I have met with no difficulties of
this nature that could not be attributed to my own carelessness, or could not be
easily avoided with sufficient patience and care. In regard to the second source of
error, I have never seen it referred to before in connection with this subject, and will
therefore speak about it somewhat further on. We should, strictly speaking, include
with the electrical measurements the subject of platinum thermometry. We shall,
however, include it with the thermal constants and treat it entirely from that point
of view.

The measurement of temperature by the change in resistance of a platinum wire
has been carefully studied by Professor Callendar, and his work is already too well
known to make it necessary for me to dwell on the fundamental part of it. His
introduction of the idea of a platinum temperature which depends on the term,

BETWEEN THE FREEZING AND BOILING-POINTS.

183

pt =

T> _ p

-^100 ■Lto

x 100,

where R0, R100 and R/ are the measures of the resistance of any one particular sample
of platinum wire at 0°, 100°, or at a temperature t, has been now almost universally
accepted.

The reduction of the platinum temperature to the air-scale was obtained from a
series of comparisons with the nitrogen air thermometer at three fixed points 0°, 100°
and 444°, which led to the well-known parabolic formula,

Too72 “ too)’

where t is the air temperature, and § a constant depending on the purity of the
platinum wire, the same for any particular purity of wire.

In selecting the wire for use in the present measurements, I was exceedingly
fortunate in possessing a sample of the original wire standardized by Professor
Callendar and Mr. Griffiths, who found its S equal to l-50.

The chief difficulty in selecting a form of thermometer for use in the calorimeter lay
in choosing a size of bulb which would give a sufficiently large change in resistance for
the rise of temperature produced in the water. On a rise of temperature of 10c, it
was necessary to be sure of the measurement to '001°, and to obtain the readings to
•0001° to have them comparable with the accuracy of the other measurements. At
the same time it was impossible to have the bulbs too long, as it introduced
increased possibilities of error in the outflow-tube of the calorimeter. For the size
of wire used (T5 millim.), and the size of the units in the resistance-box for
compensating the change in resistance, it was necessary to use about 4 metres of wire
for each thermometer.

The first thermometers made were from some of the original sample of wire, which
had been silk covered. Four metres of this wire were coiled up into a bulb, about
6 centims. long, and half a centim. in diameter, which served the purpose very well.
Two sets of thermometers were made this way at different times, and will be
described further on. The chief difficulty with this form was that, after bending
into the coil, the wire could not be annealed well enough. Annealing for a length of
time at 150° C. served to give fairly steady results. The difficulty, caused by the
exciting current, of heating in the interior of the coiled wire, was also a serious
question, which had to be carefully considered.

A pair of thermometers was made for the first tests with the water calorimeter,
which were in the usual form of bare-wire wound on a mica frame. To keep the
length of bulb within reasonable bounds, it was necessary to have these thermometers
only one-half as sensitive as the others. However, this form was far preferable to
the other, so that to produce the same sensitiveness as was required, with the most

184

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

convenient length of bulb, a smaller wire was adopted than is usually employed for
thermometric work. A thermometer was finally obtained which gave the required
sensitiveness, and had a length and diameter of bulb quite suitable for the
calorimeter. In point of steadiness and accuracy, the two thermometers forming the
differential pair made in this way could hardly be surpassed.

I propose to describe the tests made on the various thermometers used during the
course of this series of experiments. Before doing so, I must briefly describe the
resistance boxes and method used for compensating the change in resistance in the
wire due to a change of temperature.

The general plan of Wheatstone’s bridge connections for the thermometer-circuit
is already familiar. The wires leading to the bulb of the thermometer are
compensated for a change in resistance due to a change in temperature by similar
wires placed side by side with them, but connected to the opposite arms of the
bridge circuit. The change in resistance in the thermometer is compensated by
resistance coils on an opposite arm of the bridge, and a final adjustment made on a
short bridge-wire, of which the coils are suitable multiples. A change in resistance
is referred to a change in units of the box, rather than measured in ohms. It is
evident that a change in the temperature of the resistance coils, while compensating
a change in resistance in the thermometer, will produce an apparent change in the
thermometer reading. This can be corrected for either by taking the temperature of
the coils in air, or by immersing them in oil at a constant temperature. For very
accurate work, however, it is better to introduce a different arrangement. If each
resistance coil on the bridge is wound with another coil, which has the same
temperature coefficient, but a different specific resistance, then if these second coils
are connected with an opposite arm of the bridge system, any change in temperature
of the bridge coils cannot affect the balance point on the bridge wire. This method,
which was devised by Professor Callendar, works exceedingly well.

Through the kindness of Professor Callendar I have had the use of such a
compensated resistance box throughout the greater part of my measurements. This
box was exhibited to the Poyal Society in June, 1893, by Professor Callendar.
Besides the compensated resistance coils, the special features of this box are the
bridge-wire scale, which has a compensating device for a change in length due to a
change in temperature, so that the galvanometer contact point always reads at- the
same point on the scale, and mercury cup contacts for each set of coils. The
resistance coils were multiples of the bridge-wire, commencing from the smallest coil,
which was equivalent to 10 centims. of bridge-wire, and doubling always as the coils
became larger, i.e., 10, 20, 40, 80, 160, Ac., up to 2580. The resistance of the bridge-
wire was '0088 ohm per centim., so that the ten coil was rather less than T ohm.
The bridge-wire scale was of brass, very carefully divided to half-millims.., and
supplied with a vernier with lens reading to '01 millim. The total length of bridge-

BETWEEN THE FREEZING AND BOILING-POINTS.

185

wire was 40 centims., but it could be read only between 6 and 34 centims., leaving a
margin of 6 centims. at each end.

Professor Callendar was kind enough to allow me to make a resistance box after
this design. This box I have used in my later determinations of the specific heat.
It differs from Professor Callendar’s box in having a slightly greater unit, i.e.,

1 centim. of bridge-wire equal to ‘0095 ohm, and the coils were made from bare wire
wound on mica frames and annealed. Solder joins were avoided between the wire
forming the resistance coils and the copper connecting wire, by fusing directly to the
copper.

Each of the larger coils, before putting in place in the box, was tested for
compensation in a specially constructed oil-bath, the temperature of which could be
changed quickly at will in a way similar to the paraffin-bath used in the standard
resistance determinations. Each coil was also made of either two or three wires in
parallel, ‘15 mihim. in diameter, so as to avoid current heating. They were specially
designed for immersion in oil when in place in the box, but this was not found necessary.
It was not deemed necessary to test the small coils, from 10 to 40, for compensation,
as the test of the larger coils showed that the calculation of the lengths of wire
necessary was so nearly correct as to leave little room for error in the smaller coils
over a wide range of temperature. The ratio coils in the box were made from
'15-millim. platinum-silver wire wound in parallel on a mica frame, and were adjusted
to equality on the Thomson- Yarley slide box. The resistance coils were connected to
mercury cups and short-circuited when not in use by thick copper connectors.

The calibration of these boxes consists in determining the errors in the different
box coils and the calibration of the bridge-wire and scale.

In determining the total change in resistance of the thermometers between 0° and
100°, which is termed the fundamental interval, or briefly F.I., the largest that it
was necessary to use was coil 640. It is evident that, provided this coil is accurately
compensated, it is the best one to which to refer the F.I. It is entirely unnecessary
to know its absolute value in ohms provided we assume it equal to 640 even units,
and refer the other coils, including the bridge-wire, to it.

From 640 down every coil differs from the sum of all the rest by very nearly
10 centims. of bridge-wire, or the size of the smallest coil. If we compare the lengths
of bridge-wire obtained by differencing the coils in this way, we obtain the usual
series of equations of the form

640 — sumq = aY ; 320 — sunn = a% ; 160 — sum, = n3, &c.,

where oq, c<2, and a3 are very nearly 10 centims. and involve the coil errors.

If we eliminate the sum from any two equations, remembering that the next
lowest sum differs from the one before by the lesser coil, then we have a series of the
form

2 B

VOL. CXCIX. — A.

186

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

640 — 2 X 320 = ax — a2 ; 320 — 2 X 160 = a.2 — «3, &c.,
which should equal 0 if al — a.2 = <x3.

If we let the error in 640 be equal to 0, then the error in 320 = jr (al — a.2) in
terms of 640 even units,

160 = \ — a.2) — ct2 — a3},

and so on for all the coils.

The error in the bridge-wire, which we will call the bw. correction, is determined
from the error in coil 10 obtained in terms of 640 even units. The calibration of the
bridge-wire was done by inserting a small resistance, equal to about 3 centims. of
bridge-wire, into the bridge circuit, so that by short-circuiting it by a heavy copper
connector placed in mercury cups, the bridge-wire reading could be shifted the same
amount at any part of the wire. The reading was found to vary '0005 centim. per
centimetre on either side of the middle point, 19, in such a way as to increase
towards 30 and decrease towards 0. This showed that the wire was slightly smaller
towards the zero end, and lienee its resistance greater. As the equivalent length of
10 centims., obtained in the calibration of the box coils, never occurred at exactly the
same spot on the bridge-wire, there is a small correction to be applied to the values
of «l3 «o, and «3, due to their position. The correction is worked out so as to reduce
the values to a length of bridge- wire extending over the middle point, between 14
and 24. The correction is very small, however, and would produce no appreciable
error to the results if neglected altogether. In my own box the agreement of the
equivalent length for the 10 coil above and below the middle point of the bridge- wire
caused me to neglect this correction altogether.

In Table XII. I give a complete series of readings taken to determine the coil
corrections in the first box. In Table XIII. a summary of tests is given extending
over a period of a year.

Table XIV. contains the same obtained for my own box. The corrections in this
latter case are somewhat larger. The reason being that it was more difficult to
adjust the coils exactly when fused joins were used instead of solder, and at the
same time preserve complete compensation. My aim was to be sure of having this
latter condition fulfilled at the expense of the former, as the coil correction is always
a definite and measurable quantity, and easy to apply.

The signs are affixed to the corrections in the way they should be applied to the
reading. The bridge- wire correction is given per centimeter of length. In taking
the readings the galvanometer was used which has already been referred to in
connection with the comparison of the standard resistances. The sensitiveness was

obtained so as to give from 40 to 50 scale divisions per millimetre of bridge-wire on

#

reversing the current. For the small coils an external resistance of 350 ohms was

BETWEEN THE FREEZING AND BOILING-POINTS.

187

Table XIT. — Set of Readings for determining Box Coil Corrections.

Coils.

640

-sum.

32C

-sum.

160

-sum.

80-

sum.

40-

sum.

20-

sum.

10-

sum.

Reading of bridge-

r

24

•979

21

•970

20

645

19

711

25

•113

27

640

28

787

wire . . <

14

•950

12

•096

10

631

9

815

15

•135

17

624

18

911

Equivalent length

.

10

•029

9

■874

10

014

9

896

9

•978

10

016

9

876

Correction to mean

bridge-wire

0

+

•002

+

003

+

004

—

•001

—

002

-

004

Corrected length .

10

•029

9

•876

10

017

9

900

9

•977

10

014

9

872

Differences, 610-2

x 320, &c. .

+

•153

—

141

+

117

—

•077

—

037

+

142

Correction in terms of 640 coil

0

—

•077

+

032

—

043

+

•017

+

027

—

057

Table XIII. — Box Coil Corrections in Terms of 640 even Units. Box 1.

Date.

320.

160.

80.

40.

20.

10.

Bridge- wire. ;

1898.

May 6 th

- -087

+ -027

- -038

+ -021

+ -034

- -050

+ -008

„ 21st .

- -072

+ -028

- -038

+ -018

+ -026

- ’055

+ -007

„ 25th . .

- -077

+ -027

- -036

+ -017

+ -028

- -051

+ -007

1899.

J anuary 7 th .

- '055

+ -033

- -044

+ -019

+ -030

- -051

+ -007

„ 9th .

- -063

+ -040

- -042

+ -020

+ -030

- -050

+ -007

„ 12th .

- -069

+ -029

- • 045

+ -019

+ -024

- -055

+ -008

April 27th .

- -077

+ -032

- -043

+ -017

+ -027

- -057

+ -007

Means . . .

- -071

+ -030

- -040

+ -019

+ -029

- -053

+ -007

Table XIY. — Box Coil Corrections in Terms of 640 even Units. Box 2.

Date.

320. 160.

80.

40.

20.

10.

Bridge- wire.

1900.

February 10th
„ ^ 13th

„ 26th

-•029 - -054

-•025 - -048

-•030 - -053

+ -210
+ -210
+ -208

+ -144
+ -149
+ 149

- • 033

- -030

- ;031

- -074

- -071

- -072

+ -0066
+ -0061
+ -0062

Means . . .

- -028 - -052

1

+ -210

+ -148

- -032

- -072

+ -0064

2 b 2

188

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

required, which was reduced gradually to 150 ohms for the 640 coil in order to
j)reserve the same sensitiveness throughout the test, with one accumulator. The
galvanometer contact was arranged so that it could be held in contact with the
bridge-wire. Therefore instead of obtaining an exact balance and reading the
vernier, the contact was placed to the nearest millimetre or half-millimetre mark on
the scale, with the help of the vernier, and the deflection of the galvanometer
recorded. Accurate account was always kept, by repeated verification, of the
sensitiveness of the galvanometer, which never altered as much as one scale division
due to external disturbances.

For the sake of convenience, the diagram of the complete thermometer circuit is
given in fig. 6. This shows the relative position of the resistance and compensating
coils in the bridge system, the position of the ratio coils and bridge-wire. When

differential thermometers are used we have them connected on opposite arms of the
bridge, at P and C, and arranged so that the compensating leads for thermometer P
are in series with thermometer C, and the compensating leads for C connected with
P. Where P and C are at the same temperature, and of the same resistance, it is
evident that the bridge system is in equilibrium with the galvanometer contact at
the middle of the bridge-wire. For a change in the temperature of either P or C the
bridge reading shifted either to the right or left, and when too great to be read on
the wire, was compensated by the resistance coils. A change of temperature in C
higher than P, however, could not be recorded beyond the bridge-wire. It was
therefore necessary to arrange that P should always be used for measuring a change
in temperature higher than C. The resistance coils were brought into the circuit by
removing the heavy copper contacts from the mercury cups. When these contacts

BETWEEN THE FREEZING AND BOILING-POINTS.

189

were removed, the contacts of each corresponding compensating coil were removed at
the same time. To obtain the fundamental constant (R100 — R0) or interval, F.I.,
both C and P are balanced when immersed in melting ice, and then with C in ice and
P in steam. To obtain the difference between the intervals of the two thermometers
both are read when in steam. This gives the data required for converting into
degrees a change of resistance in P relative to C.

During the progress of the present experiments, five pairs of differential thermo¬
meters were made and tested. In describing these, I shall letter them A, B, C, D,
and E, respectively. The thermometers of pair A were made of the original silk-
covered •15-millim. platinum- wire, about 25 ohms resistance each. The bulb of each
thermometer was about 6 centims. long and was fastened by solder joins to flexible
copper leads placed side by side with compensating leads. The protecting tube was
of glass, about 25 centims. long and a few millims. in diameter. The ends of the
compensating leads near the bulb were connected by a small piece of platinum-wire
about 4 centims. long. This was to correct any conduction error on the wire in the
thermometers by heat conduction from the leads. This device was also used for all
the other thermometers.

Thermometer B was made of 12|~ohms resistance, or one-lialf the sensitiveness of
the other pair. Each thermometer of the pair was wound in the usual way on a
mica frame, from the 6-millim. bare platinum-wire, and annealed at a low red heat.
As these thermometers proved eventually to be too bulky for convenience in the
calorimeter, they were soon discarded. It will, therefore, not be necessary to give
them further mention.

Thermometer C was made from a pair of silk thermometers similar to A. The
platinum-wire was fused to copper-wires, which in turn were soldered to copper leads.
These thermometers proved satisfactory in many ways, although they finally gave
trouble from defective insulation and had to be abandoned. These thermometers
were used in our first preliminary measurements of J during the summer of 1898.

Thermometer D was the first pair made from ‘10-millim. platinum- wire. This wire
was some sent out to Mr. R. O. King by the Cambridge Scientific Instrument
Company. Its 8 was given as R50, which was subsequently verified by Mr. Tory in
the course of his work. Each thermometer was about 20 ohms, in resistance, and
was made by winding on a mica frame. The bulb was about 5 centims. long and
between 6 and 7 millims. in diameter. Owing to the inconveniently small F.I. of
this pair of thermometers (about 700 units of the box instead of 1000 for 100° C.), it
was supplanted by pair E.

Thermometer E is by far the most important pair, as with it all the later measure¬
ments of the specific heat were obtained. The wire used in- making each thermometer
of the pair was drawn down to HO millim. from the original 6-millim. platinum- wire.
The resistance of the thermometers was about 25 ohms each, and gave a F.I. about
970 units of the box. The bulbs were about 7 millims. in diameter and about

190

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

6i|- centims. long, and were made, as in thermometer D, of the bare wire wound on a
mica frame. The first arrangement was with the platinum- wire fused to about No. 18
copper-wire, which in turn was soldered to the copper leads about 6 centims. above
the bulb. This was changed to having the wire fused to much longer copper- wires,
which were soldered to the leads at a point considerably beyond the glass-tubes con¬
taining the bulbs. This avoided the changing of the temperature of the solder-joins
in the glass-tube. The final arrangement was to have the wire gold-soldered to heavy
platinum-wires, which in turn were fused to copper-wires about 6 centims. above the
bulbs. These wires were then soldered to the main leads at a point sufficiently
beyond the glass-tube so as to remain unaffected by a change in temperature in the
interior of the glass-tube. All these changes were made to improve the thermometers,
although the last one was not really necessary. A very considerable uncertainty was
introduced with the first arrangement, which was removed on removing the solder-
joins from the interior of the thermometer-tubes.

The TO-millim. wire is exceedingly delicate to use for thermometric work, and great
care had to be exercised in constructing the thermometers and in handling them.
They gave, however, exceedingly consistent results. As a check, a sample of the
wire was given to Mr. Tory, who very kindly determined its 8 by comparing it with
a piece of the original •15-millim. wire. This came out P502, and showed that no
change had been produced in the 8, due to its having been drawn from the
larger size.

The fixed points 0° and 100° upon which the accuracy of the F.I. depends, were
obtained as usual with a mixture of finely-cracked ice and water, and the usual
form of hypsometer. In regard to the constancy of these points, the former depends
on the percentage of ice or water present in the mixture, and its rate of melting,
while the latter depends on the accuracy of reading of the barometer, accepting in
both cases the purity of the ice or water. Great care was always taken with the
freezing-point mixture, to have it compact and firmly placed in a copper vessel,
heavily lagged, and in which water could be made to circulate through the ice around
the thermometer bulbs. The thermometers were as far as possible placed side by
side, separated only by a thin partition of ice.

After obtaining the balance-point with both thermometers in ice, one, P, was removed
to the steam-jacket, leaving the other, C, still in ice. The change in resistance in P
was compensated by the resistance-coils until the reading was brought on to the
bridge-wire. When a sufficient time was allowed (about 15 minutes was generally
sufficient) for the attainment of a steady temperature, as shown by the steadiness of
the balance-point, the reading of the bridge-wire was recorded. Thermometer P was
then returned to the ice-bath, and the first reading repeated, which gave a measure
of any change of zero in P. The sensitiveness of the galvanometer changed slightly
between the two points owing to the increase in resistance in the two arms of the
bridge system, but this was determined always at both points.

BETWEEN THE FREEZING AND BOILING-POINTS.

191

In the tests to be described, an external resistance of at least 40 ohms was inserted
in the battery circuit, which was supplied from one accumulator. The current was
never as much as ’02 ampere in each thermometer, and current-heating could he
safely neglected. In view, however, of a possible elfect of current-heating on the
differential readings, the F.I. of thermometer D was determined for different external
resistances, but no effect could be measured. The current was left continuously
running during a test, and was always the same as that used in the calorimetric
experiments.

Three Fortin barometers, supplied by Elliott Bros., of London, were used to
obtain the steam-point. They were Nos. 571, 572, and 573. They had all been filled
originally by Elliott. During the preliminary part of the experiments, barometer 572
was used, but owing to an accident was replaced later by 571. Barometer 571 was,
however, re-filled later as a check, by boiling out with mercury, and was compared
with 573, to which most of the later steam readings were referred. As a check also
the scale of barometer 572 was verified, and found correct to within T millim. This
was of sufficient accuracy for the determination of the F.I., as the mercury height
could be measured accurately only to T millim. with the vernier supplied with the
scale. It was possible to estimate to '02 millim. with a little care.

A comparison of 572 re-filled with 573, made on February 26, 1900, is given below.

No. 573, 75'415 centims., t = 17*5; No. 572, 75'410 centims., t = 17'3.

Second setting,

No. 573, 75'415 centims. ; No. 572, 75'400 centims.

/

The temperature of the mercury column was taken from a thermometer embedded
in the barometer case. I decided to adopt the readings of the highest barometer
as likely to be most accurate. It is probable that barometer 573 was correct to
'01 centim. in its readings over different dates. An error of '01 centim. in setting
and reading the height of mercury would produce an error of about '004° on the
steam point, which is about the order of agreement given in the measurement of
the F.I. from time to time. In repeating readings of the F.I.. where it was not
necessary to alter the setting of the barometer, much closer agreement than this was
attained. The barometer readings were reduced to 0° C. and latitude 45°, and
corrected for temperature by the formula

H0 = H (1 - -0001614?:) (1 - -000033) + -0002£ centim.

The essential scheme followed for the determination of the F.I. has already been
laid down. The following tables contain the results of the tests made on the
different instruments used in these experiments. Owing to the importance attached
to thermometer E, it is given first place. For this thermometer it will be necessary
to divide the tests into four groups.

192

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

Group 1. contains the results when the '10-millim. platinum-wire is fused to short
copper- wires, which in turn are soldered to the leads inside the glass-tubes of the
thermometers, so that the solder-joins are subject to the temperature change
0° to 100°.

Group II. gives the tests when the solder-joins inside the glass-tubes were done
away with.

Group III. contains the results in the case when the •10-millim. wire was gold-
soldered to heavy platinum-wire, which was fused to the copper leads.

Group 1Y. contains the determination of F.I. in terms of the second box. It comes
smaller owing to the resistance of the unit being greater than in the first box.

In Group I. the variation from the mean value is somewhat larger than the
errors of reading the barometer, amounting in the extreme case to '007°. This was
attributed to an uncertainty in the solder joins.

Group I. — Coils 640 + 320 -j- 10 = 969‘876 units. Thermometer E.

10 units = 1° C. nearly. Box 1.

Date.

Bridge-wire

corrected.

Total units.

Barometer

corrected.

F.I. corrected to
76-00 eentims.

February 24, 1899 ....

+ 2-964

972-840

76-196

97-2141

5? ....

+ 4-597

974-473

76-630

97-2217

„ 27 „ ....

-1-396

968-480

74-938

97-2260

,, 28 „ ....

-2-022

967-854

74-789

97-2162

„ 28 „ . . . .

-2-399

967-477

74-683

97-2160

Mean . . .

. . . 97-2188

Group II. — Coils 640 + 320 -f- 10 = 969 '8 76 units. Thermometer E.

10 units = 1° C. nearly. Box 1.

Date.

Bridge- wire
corrected.

Total units.

Barometer

corrected.

F.I. corrected to
76-00 eentims.

May 15, 1899 .

+ 3-006

972-882

75 • 9S5

97-2935

„ 15 „ .

+ 2-495

972-371

75-844

97-2926

,, 18 „ .

+ 1-669

971 -545

75-616

97-2916

August 10, 1899 .

+ 1-302

971-178

75-492

97-2991

„ 10 . .

+ 1-279

971-155

75-485

97-2993

September 11, 1899. . . .

+ 2-320

972-196

75-792

97-2939

„ 11 „ . . . .

+ 2-211

972-087

75-760

97-2944

„ 21 „ . . . .

+ 0-670

970-546

75-314

97-2993

Mean . . .

. . . 97-2955

BETWEEN THE FREEZING AND BOILING-POINTS.

193

In Group II. the agreement is much better, and it can hardly be said that the
variations can be attributed to other causes than the setting of the barometer. For
tests made on the same date, where the interval was repeated with but a small
change in barometric reading, which could be followed by the vernier without other
adjustments, the readings are as a rule exceedingly consistent.

The two tests in Group III. are given in full.

Group III. Date, September 26, 1899.

First determination.

Both in ice, reading of bridge- wire .
C in ice ; P in steam .

Second determination.

Both in ice, reading of bridge- wire .
C in ice ; P in steam . . .

Both in steam .

23-173. No coils.

23-757 + coils, 640 + 320 -f 10.

23-175, 23-172. No coils.

23'646 -f- cods, 640 320 -j- 10.

25-143. No coils.

Barometer in first determinations. Uncorrected, 7 5 "073 centims. at temp. 18°"9 ;
corrected, 74"851.

Barometer in second determinations. Uncorrected, 7 5 "092 centims. at temp. 18°"8 ;
corrected, 74 '870.

In first determination. In second determination.

Bridge-wire .... — 0"584 — 0‘473

Bw. correction .... 4 3

- 0-588 — 0-476

Coils . 969-876 969*876

969"288 in box units. 969"400 in box units.
Barometer correction . -}- "4093° + "4025°

F.1 . 97 ‘3381 in degrees. 97 "3425 in degrees.

Meanvalue^. . . . 97"3403.

Group IV.— Coils 640 + 160 + 80 + 20 = 900-128 units.
9 units = 1° C nearly. Box 2.

Date.

Bridge-wire

corrected.

Total units.

Barometer

corrected.

F.I. corrected to

7 6 '00 centims.

February 14, 1900 ....

+ 4-143

904-271

75-778

90-5009

„ 14 „ . . . .

+ 2-195

902-323

75-191

90-5006

>> 16 ,, . . . .

+ 4-307

904-435

75-816

90-5037

ti 16 .

+ 4-328

904-456

75-838

90-4994

» 16 .

+ 4-361

904-489

75-846

90-5002

VOL. CXCIX. — A. 2 C

194

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

Summarizing we have the F.I.’s of thermometer E, as measured by the two boxes,
as follows : —

Group I. — 97 '21 88 measured with 1st box.

55 55 55

55 55 55

„ 2nd „

This gives the corrections to be applied to the readings to make 100 10-units
equal to 100° on the platinum scale :

Group 1. — 2'8610 per cent. Group III. — 27324 per cent.

„ II.— 27795 „ .„ IV.— 10-4962

The difference in the corrections, with the exception of Group IV., is due to the
change in the lengths of the thermometer wire in changing the leads. No change
was made to the thermometers themselves between Groups III. and IV.

The various tests on the other thermometers are given now as under. With
thermometer D no difference could be noted in the F.I. measured with 80 ohms
in the external circuit, or with 50 ohms. The tests with thermometer A are
important as illustrating the results to be obtained from silk-covered wire thermo¬
meters, and also as they were the thermometers used in the mercury experiments.

Thermometer A. Box 1.

Date. F.I. corrected.

May 22, 1897 . . . 99 '9327

„ 24 „ . 99'9245

II. — 97'2955

III. — 97-3403

IV. — 90-5009

With thermometer C, also silk-covered thermometers, the tests are not so
satisfactory.

Thermometer C.

Date.

F.I. corrected.

Date.

F.I. corrected.

April 30, 1898 .

. . 101-4031

July 11, 1898 .

•

101-3908

May 2 „ .

. . 101-3931

August 8 ,,

• -- — ---

101-4006

BETWEEN THE FREEZING AND BOILING-POINTS.

195

Thermometer D. Box 1.

Coils 640 + 160 = 800'030 units.
8 units = 1° C., nearly.
With 80 ohms external circuit.

Date.

Bridge-wire

corrected.

Total units.

Corrected

barometer.

F.I. corrected to
76-00 centims.

July 21, 1898 .

-4-679

795-351

75-806

79-5924

-4-655

795 • 375

75-816

79-5913

)) B .

-4-614

795-416

75-826

79-5925

With 50 ohms external circuit.

11 11 .

- 4*651

795-379

75-816

79-5917

All these fundamental intervals, of course, only apply to thermometer P of each
pair, or the one which is used to determine the rise of temperature in the water.

No separate determinations are required when both the thermometers are at the
same temperature over the scale between 0° and 100°. The correction is simply to
thermometer P in its reading relative to thermometer C, when the water is heated
through so many degrees in. the outflow end of the calorimeter. The “ cold” reading
of the thermometers, when in place in the calorimeter, at any temperature of the
water-jacket, will be the differential reading of the thermometers at the temperature
indicated by the thermometer of the water-jacket. The effect of conduction from the
ends of the calorimeter will appear as a slight change in this differential reading, but
this is never more than '01° or ‘02~, and only comes in at the extremes of the range.

In regard to the errors referred to in the first of this section, to he met with in the
practical employment of platinum thermometers for very accurate work, the first one,
due to a change in zero, can always be avoided by sufficient annealing and offers no
difficulty. The second one is by far the most important, and is caused by the
conduction of heat away from the air around the bulbs through the metal leads.
This is rendered worse rather than better by the presence of the compensating leads,
because of the greater amount of good conducting material introduced into the
thermometer tubes. The employment of a small length of wire to connect the
compensating leads cannot rectify it, nor will the prevention of convection currents of
air in the glass tubes render it harmless. It can be reduced to a negligible quantity
only by immersing a sufficient length of the thermometer tube, and can always be
measured by withdrawing the thermometer tube more or less from the vessel or
steam-jacket in which it is immersed, and determining the drop in temperature

2 c 2

1 9(>

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

indicated by the bulb. I took special care to reduce this connection, which could
easily amount to ‘01 or even more in the steam-jacket, to a negligible quantity, so
that it could not have affected the F.I. as determined in my experiments to as much
as ‘00 1°, and probably much less. Convection currents in the stem of the thermo¬
meter enclosed in the glass-tube were avoided in all my thermometers by lagging the
leads down to within a few centims. of the bulb with cotton-wool.

The insulation between the leads of the thermometers could be detected very
quickly by a very simple adjustment in one of the box contacts, so that the battery
in series with the galvanometer could be made to detect at once the smallest leak
between the connecting and compensating leads.

In correcting the differential readings to the air scale two of the ordinary parabolic
formulae are combined.

If ptx and tx be the platinum and air temperatures for one thermometer of the
differential pair and pt2 and t% be the same for the other, then for one we have

and for the other

h ~ Vh =

8

' tL _ A\

TOO2 100/

U - pt%

k\
100/ ’

from which we have, for the differential reading,

(C ~ Ph) ~ (h ~ Ph) = bybbb ^ ~ + h ~ 100)-

The correction is always small, and amounts to T° in the extreme at the ends of
the scale for a difference of 8°. It vanishes altogether at 50°, changing sign at that
point.

In determining the correction ptx and pt.2 may be substituted in the right-hand
side for a first approximation. A second approximation generally gives the correction
with sufficient accuracy.

Sec. 3 d. — Measurement of Time.

The method of measuring the average rate of flow’ over the time of any experiment
was to divide the total weight of water by the time of flow. This total time of flow
was generally of the order of 15 minutes, or 900 seconds. The interval was recorded
on a chronograph, which marked seconds 1 centim. long, from a standardized clock.
The drum of the chronograph revolved at the rate of one revolution per minute, and
the record of each second vras made by a lateral kick in a continuous fine from a pen

BETWEEN THE FREEZING AND BOILING-POINTS.

197

marking an attached sheet of paper. The pen was connected to a relay, which was
excited by an electric current whenever the pendulum of the clock swung through a
globule of mercury at the middle of each stroke. The stream of water from the
calorimeter could be made to flow out of either one of two nozzles before it entered
the tared flask for obtaining its weight. The switch-over device was made from a
3-way glass tap, and so arranged that when the tap was turned so as to change the
flow from one tube to another, the time of closing one and opening into the other was
recorded by the pen on the chronograph sheet. Between the opening and closing,
the flow was shut off altogether, but as this only amounted to from two to three-
tenths of a second, the expansion of the rubber tube connections in the water circuit
prevented the flow of water through the calorimeter from being interrupted, and any
sudden shock to the conditions was avoided.

Fig. 7 gives a general plan of the switch¬
over device. The handle of the glass tap
was connected to a long arm, which could be
moved between two stops, representing the
position when either nozzle was open. At
the time of switching over, two marks were
recorded on the chronograph, the mean of
which was taken in estimating the interval
of any flow. These two marks could each be
estimated to '01 sec. very easily with a
millimetre scale, and were probably in all
cases accurate to '02 sec. on 900 seconds.

The standardization (indirectly to standard
time) of the clock marking the seconds was
done at frequent intervals during tire course
of the present series of experiments by com¬
parison with a Frodsham and Keen ship’s
chronometer. The rate of this chronometer was determined not only by direct
comparison with the standard clock at the University Observatory, but by repeated
telephonic communication between an observer at the Observatory and myself in the
Physics Building. This rate, which was a slight gain, was extraordinarily constant.
The rate of the clock, a loss, varied considerably between winter and summer, although
the variation was very consistent and regular. The rates during the corresponding
months of a year agreed almost to I second a day.

198

DR. IT. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

Table XVI. — Clock Comparisons.

1898.

1899.

May.

June.

July.

Aug.

Sept.

Oct.

Dec.

Jan.

Feb.

March.

seconds.

seconds.

seconds.

seconds.

seconds.

seconds.

seconds.

seconds.

seconds.

seconds.

1

—

28

35

35

—

38

—

—

—

—

9

—

27

32

36

—

—

—

24

—

— —

3

—

27

—

36

—

—

—

24

—

—

4

—

27

33

33

—

—

—

—

—

18

5

—

28

33

—

—

—

—

25

—

—

6

28

35

35

25

7

—

- -

33

—

—

—

—

25

20

_

8

—

—

33

35

—

—

—

—

—

18

9

—

—

36

37

—

—

—

24

—

10

—

—

—

35

—

—

—

24

—

—

11

33

35

23

_

12

—

—

35

36

—

—

—

24

—

—

13

—

—

—

35

—

—

—

23

22

14

—

—

33

—

36

—

29

—

—

15

—

—

34

—

—

—

28

—

—

—

16

34

37

29

23

17

—

—

—

—

—

—

28

—

—

—

18

—

—

33

—

—

—

—

—

—

—

19

—

—

34

—

—

—

28

23

—

—

20

—

—

32

—

—

—

27

23

—

—

21

34

39

26

23

_

22

—

22

32

—

—

—

25

—

18

—

23

—

24

35

—

37

—

26

22

—

—

24

—

30

—

—

—

—

25

—

—

19

25

—

32

34

—

—

—

23

—

—

—

26

34

24

27

—

32

34

—

—

—

25

—

—

—

28

25

32

35

—

—

—

25

—

—

i

29

26

33

35

—

—

—

25

—

—

30

26

33

34

—

—

—

26

21

—

31

27

—

—

—

—

—

25

_

—

BETWEEN THE FREEZING AND BOILING-POINTS,

199

Table XVI. — Clock Comparisons — continued.

1899.

1900.

April.

May.

June.

July.

Sept.

Oct.

Nov.

Dec.

Feb.

March.

seconds.

seconds.

seconds.

seconds.

seconds.

seconds.

seconds.

seconds.

seconds.

seconds.

90

1

9

—

—

. -

—

—

—

—

—

3

—

—

—

—

—

—

—

—

—

4

19

22

—

—

37

40

—

—

!

—

5

18

38

38

6

19

—

30

—

—

39

—

—

23

—

7

—

21

31

—

37

—

—

_

23

23

8

—

22

31

—

—

—

—

—

23

21

9

—

22

32

—

—

—

—

—

—

20

10

19

23

11

19

23

32

36

37

—

_

—

—

20

12

19

' -

31

—

—

—

—

—

—

13

19

—

29

—

37

—

39

—

—

14

—

—

32

—

—

—

36

—

—

15

33

36

20

16

—

34

—

—

—

—

—

20

17

20

—

—

—

—

35

—

—

— -

IS

20

—

34

—

—

—

—

20

19

20

—

34

—

—

—

—

—

22

—

20

20

33

19

21

—

—

34

_

_

_

_

—

—

—

22

—

- -

35

—

—

—

31

—

—

20

23

—

—

35

—

—

—

_

—

—

—

24

—

26

—

—

—

—

—

—

—

—

25

_ _

35

38

26

—

—

35

—

—

33

—

—

—

27

—

—

35

— -

—

—

33

28

—

—

28

—

—

35

—

—

—

—

—

—

—

29

—

26

34

—

—

—

28

—

—

30

—

—

—

—

—

—

—

—

31

200

DE. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

The following Table contains the results of the comparisons for the chronometer
obtained by telephonic communication : —

Table XY.

Date.

Gain.

Date.

Gain.

Jan. 3 to Jan. 4, 1899 . .

>' 4: ,, ,, 5, ,,

n ^ 5) ?) JJ *

,, 6 ,, „ 7, ,, . .

„ 7 ,, „ 9, ,,

„ 9 „ ., 10, „ . .

„ 10 „ „ 11, . .

6 • 5 secs.

6- 5 „

7- 0 „

6- 5 „

12-5 „

7- 0 „

7-0 „

Jan. 11 to Jan. 14, 1899 . .

„ 14 „ ,, 16, ,, . .

„ 16 „ „ 23, „ . .

)> 1) JO, ,,

„ 30 „ Feb. 6, ,, . . .
Feb. 6 „ „ 13, „

„ 13 ., „ 20, „ . .

19 -5 secs.
12-0 „

48 - 5 ,,

49- 0 ..

46-0 „

48-5 .,

48 ’5 „

Mean gain .... 7 ‘0 secs, in 24 hours.

From March 20 to March 21 of the same year (1899) the gain was 7' 0 secs. ; from
February 9 to February 10, 1900, the gain was 7'0 secs. ; from February 10 to
February 12 it was 15'0 secs.

Ill August, 1898, the chronometer was taken to the Observatory for two weeks, and
daily comparisons were made with the standard clock. Its rate was found to vary
between 6 and 9 seconds gain per day, which is a somewhat greater irregularity than
I obtained, although the mean value agrees very well with the later comparisons.
In rating the clock, 1 considered it safe to assume the rate of the chronometer
constant to within a second from day to day, and equal to a gain of 7 secs.

In Table X VI. , I give the comparison of the clock with the chronometer from
May, 1898, to the close of the present work. The seconds represent the loss of the
clock per day, and are not corrected for the rate of the chronometer.

In Table XVII., a summary of the previous Table is given, showing the greatest
and least loss per month, with the mean loss corrected for the chronometer, which is
obtained by subtracting 7 seconds. As far as possible, the rate of the clock was
obtained over any day in which an experiment was obtained.

BETWEEN THE FREEZING AND BOILING-POINTS.

201

Table XYII. — Summary of Clock Comparisons.

Month.

Greatest loss
in seconds.

Least loss
in seconds.

Mean loss
in seconds.

Mean loss
corrected for rate
of chronometer.

May, 1898 . . .

26

25

26

19

June, ,, ...

33

27

30

23

July, „ . . .

36

32

34

27

August, „ . . .

37

33

35

28

September ,, ...

39

36

38

31

December, ,, ...

29

23

26

19

January, 1899 . . .

25

21

23

16

February, ,, ...

22

18

20

13

March, ,, ...

19

18

19

12

April, ,, ...

20

18

19

12

May, „ . . .

23

21

22

15

June, ,, . . .

35

29

32

25

September, ,, ...

38

37

38

31

October, ,, . . .

40

38

39

32

November, ,, ...

39

31

35

28

February, 1900 . . .

23

22

23

16

March, ,, ...

23

19

21

14

The rate of loss diminishes in winter and just doubles during summer. This is
probably due to the effect of the dry furnace heat in the building during winter on
the wooden pendulum of the clock, in contrast to the more humid climate of the
summer months. The furnace fires are started about the month of November and
discontinue some time in April.

Sec. 3e, — Measurement of Weight.

In all the older methods of calorimetry, the question of evaporation of the water
becomes a serious one for consideration. In the present experiments there were no
such difficulties. The stream of water flowing from either one of the two nozzles on
the outflow end of the calorimeter was caught in a weighed flask, which was fitted
with a rubber stopper through which the nozzle passed. Through a second hole in the
stopper a tube was fastened containing calcium chloride, so that the air in the flask,
displaced by the inflowing water, passed through the calcium chloride. At no point
between the calorimeter and the interior of the flask did the water come in contact
with air.

In fig. 8 a drawing of the flask is given, showing the position of the calcium-chloride
tube. The hole through which the nozzle of the calorimeter is thrust is closed, when
the flask is removed, by a glass stopper. A similar stopper closes the end of the
calcium-chloride tube and prevents the absorption of water vapour from the air.

The capacity of each of the two flasks, which were used in the experiments, was
about 750 cub. centims., but the amount of water weighed in them was never more

VOL. CXOIX. — A. 2 D

202

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

than 650 cub. centims., and usually varied from 350 to 600 grammes. The weight of
the flasks was taken empty, just previous to a determination of the flow and after
they had stood for some time unstoppered inside the balance case. The interior of the

flasks was always wet on account of the water
which had been weighed in them on a previous
occasion.

The weight was taken on a large Oertling
balance, which proved to be most suitable for the
purpose. It was sensitive to less than 1 milli¬
gramme with 500 grammes, which gave a measure
of the weight to a sufficient accuracy. The
weights used were brass, and were one of several
sets supplied us by Oertling. They were
kept entirely for the present purpose. It is
exceedingly unlikely that their errors would
amount to as much as 1 milligramme, especially
as the several sets sent us by Oertling agreed
much closer than that, and the different weights
in the same set agreed very closely amongst themselves. Even if it could be imagined
that the sum of the errors of the weights used in any given weighing could have
amounted to 10 milligrammes, that would have produced an error in the estimation of
the flow of only 1 part in 50,000, whereas it is most probable that the error was not
so much as a tenth of this.

The correction to be applied to the weight for the ratio of the arms of the balance
was found to be less than 1 in 100,000 for the weights used.

In the reduction of the weight of the water in the flask to weight in vacuum, it is
necessary to correct for the presence of water- vapour in the displaced air. This water-
vapour is retained in the calcium-chloride tube when the air is driven out and
therefore appears not only in the weight of the flask empty, but when it contains the
weight of water. It is consequently eliminated from the final weight. The actual
weight of air displaced, however, is less than it otherwise wrould be, by the presence
of the water-vapour. In applying this correction it may be assumed that the air inside
the flask is completely saturated with water-vapour at the temperature of weighing.

Where brass weights are used the ratio of the weight in vacuum to the weight in
air is given by the expression

W vac. _ / \t_ \t \

W air ~ \ + S “ 8 4/’

where for the calculation of X t we can use the formula

\ _ _ \ _ P P

1 ~ (1 +’-003660 T ‘

BETWEEN THE FREEZING AND BOILING-POINTS.

203

In this case \0 = ’001293 ; P = 760 ; p = the observed barometric pressure at the
time of weighing ; p = the vapour-tension of water at the temperature of weighing.

This latter correction for the water-vapour is small and amounts to about 2 parts in
100,000 on the flow at the usual room temperature. In applying this correction, it
was necessary to have the water enter the flasks at the temperature of the room,
which was very nearly the temperature of the balance case. This was important,
especially when the calorimeter was at a temperature very different to the room
temperature. A cooler, consisting of a bath through which a constant stream of water
could be made to flow, was arranged adjacent to the calorimeter, so that the outflowing
water from the calorimeter was passed through a spiral of copper tube immersed in
the water before passing through the switch-over device into the flasks. The tem¬
perature of the cooler was maintained near the temperature of the room by controlling
the temperature of the stream of water by a gas flame. A stirrer wTas also fitted up
for the bath.

A small change in temperature of the cooler, during the time of an experiment,
required a small correction to the flow. This depended only on the readings of a
thermometer in the cooler- water just previous to the switching over of the flow into
the weighed flask, and just after it was turned off from the flask at the end of the
interval. If dt represents the change in temperature obtained from the two readings,
v the weight of water filling the copper-spiral in the cooler, and - a the coefficient of
expansion of water, then the correction to be applied to the flow is a vdt. When dt is
of the order of a degree, this correction is just negligible for the volume of the total
length of copper-tube used, which contained about 22 grammes of water.

Sec. 4. — Description of the Apparatus and Method of Making the Experiments.

The Calorimeter. — A general plan of the calorimeter is shown in fig. 1 (p. 153). The
first three calorimeters were made at Bonn, and sent out to the laboratory unexhausted.
We exhausted the vacuum-chamber of two of these, but the third one was not used
owung to the adoption in later experiments of a slightly different design. They all
had the same dimensions, with a fine-bore flow-tube 2 millims. inside diameter and
50 centims. long, which was fused at both ends to larger tubes 25 centims. long and
1’8 centim. inside diameter. These larger tubes wmre sealed into the vacuum-jacket
made from a glass-tube 4 centims. in diameter. The seal at each end was made at
about the middle of the larger flow-tubes, at a point about 1 1 centims. from the end
where the fine-bore tube was sealed on. Two side tubes were sealed into the larger
tubes at each end, but were eventually done away wfith in the later design, with the
exception of one on the inflow end. The vacuum-jacket was exhausted on a large
five-fall Sprengel pump with a McLeod gauge for determining the vacuum, and con¬
nected to the pump by a side tube fused into the glass of the chamber. When
exhausted sufficiently, to a vacuum of about ’002 millim. as shown by the gauge, the

2 d 2

204

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

connecting tube was fused together on the pump so that the vacuum chamber was
permanently and hermetically sealed. The jacket was carefully heated during
exhaustion so as to drive off water-vapour and occluded gas from the glass.

Three other calorimeters have been made by Eimer and Amend of New York,
since September, 1898. The vacuum jackets of these were all exhausted by them
while heating the calorimeter in an asbestos oven. These calorimeters were of
a similar design to the earlier ones, except in having only one side tube at the inflow
end, and in having fine-bore flow tubes of different sizes, ranging from 2 millims. to
a little over 3 millims. One of the calorimeters was prepared with phosphorus
pentoxide in the vacuum chamber, but this proved to be rather a drawback than
otherwise, because of the greater capacity for heat of the calorimeter introduced by
the P305. It was essential to have the jacket very perfectly exhausted to avoid the
heat-loss due to convection currents of air, which acted in such a way as to make the
radiation loss appear large and uncertain. Small losses, however, from conduction
and convection in the residual vapours in the jacket produced no errors on account of
the steady temperature conditions during an experiment. The radiation loss would
depend on the state of the glass surface, but would apparently be increased at the
lower temperatures, after the calorimeter had been maintained for several hours
during an experiment at one of the higher points. It was impossible always to count
on the constancy of the heat-loss from one experiment to another, even with one
calorimeter at the same temperature, as it depended so much on the past history.
The change in heat-loss occasioned by the driving-off of occluded gases and vapours
from the glass when the calorimeter was at a high temperature took place so slowly
that, during an experiment extending over several hours, no measurable alteration in
it could be noticed. This same effect of a slight change in heat-loss from time to
time was also noted in the calorimeter used for the mercury experiments.

In fig. 9 is given a cross-section of the calorimeter and interior fittings in place, in
the water-jacket. The thermometer bulbs are shown included in their glass cases.
These cases were about 9 millims. in . diameter, and extended the full length and
a little beyond the ends of the outflow and inflow tubes. The ends of the thermo¬
meter cases over the bulbs were enclosed in thin copper cylinders, gold-plated, about
10 centims. long. These copper cylinders served the double purpose of preventing
generation of heat in the vicinity of the thermometer bulbs by the electric heating
current, and of helping to equalize the temperature of the water around the bulbs.
The heating current was conveyed to the copper cylinders by eight No. 12 copper
wires at each end, which were soldered into slits cut for them in the ends. The
cylinders were made with closed ends, in one of which a hole was made for soldering
in the platinum heating wire, and in the other a special screw clutch for catching the
wire after the cylinders were put in place in the calorimeter. Sections of the
cylinders are given in fig. 10.

The central heating wire for the fine-bore tube was made in three ways ; either

BETWEEN THE FREEZING AND BOILING-POINTS.

205

a central solid platinum wire about '4 millims. in diameter, or six strands made up
of 6-millim. platinum wire in parallel, or a flat wire twisted into a spiral down the
fine tube. Of these the first proved to be the most satisfactory and gave the steadiest
results when held central in a tube by a silk-covered rubber cord wound around it in
spirals of about 1 centim. The stranded heating-wire, although excellent, was more
difficult to handle, especially in putting the fittings of the calorimeter together. The
flat wire did not require to have the rubber band wound around it, and it was of the
full width of the tube. It was very difficult indeed to arrange the interior of the

calorimeter when using this form of twisted heating-wire, as a small strain on the
wire was almost sure to tear it apart. Moreover, in all experiments such as these,
where water is heated by an electric current conveyed in a wire, the temperature of
the wire is always above that of the water, so that as this form of wire touched the
sides of the glass-tube the heat-loss was increased by the heating of the glass-tube at
the points where it was in contact with the wire. Still, this arrangement gave
excellent results for two calorimeters in which it was tried, and served as a satisfactory
check on the results obtained by the other heating-wires. The solid heating-wire
was finally adopted as offering the least mechanical difficulties.

206

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

In fitting the copper cylinders into place, they were each wound with a small
rubber tube, shown in section, in fig. 9. Through these rubber tubes on the two
cylinders small copper wires were placed, and soldered to the ends near the heating-
wire. These copper wires served both as potential terminals as well as a method of
holding the rubber tubes in place on the copper cylinders. The rubber tube served
three purposes ; the holding of the cylinders central and firm, the stirring of the
water as it flowed around the thermometer bulbs, and the insulation of the potential
leads.

The platinum heating- wire was fused at one end to a copper wire of the same size,
which was in turn soldered with pure tin into the hole in the end of cylinder B,
fig. 10. The other end of the heating-wire was soldered with tin directly to three
copper wires, which served to draw the platinum wire into place in the fine-bore
tube.

When cylinder B, which was placed in the outflow end of the calorimeter, was
shoved down into place as far as it would go, the heating-wire was about 3 centims.
shorter than the fine tube at the further or inflow end. The three copper wires,
which were attached to the heating- wire and protruded from the calorimeter tube,
were pushed through slits cut for them between the copper screw and nut (shown in
fig. 10) on cylinder A. These three wires could be drawn through readily with the
screw only partly in place, and in this way by pulling the wires through, cylinder A
was shoved down the calorimeter tube into its place in the inflow tube. When in
place a screw-driver was inserted, and a jack to hold the cylinder from turning, and
the copper screw turned into place. When it reached the part of the thread where
the slits ended, the three wires were firmly gripped between the thread and screw.
The copper screw was smoothed on the end so as not to cut through the wires, but
simply to jamb them against the screw thread. When sufficiently firm a specially
constructed cutter was introduced, and the wires cut off just at the head of the
copper screw, this left the cylinder firmly attached to the heating- wire by the three
copper wires. The glass-tubes for the differential thermometers were placed in the
two ends of the calorimeter, and slid into the two cylinders prepared for them. The
tubes were put in empty, as it was found better to introduce the thermometers
themselves after the calorimeter was fitted up and in place in the water-jacket. The
ends of the calorimeter were closed watertight by means of a rubber stopper placed
on each glass thermometer tube near the end of the calorimeter tube. The eight-
copper wires at the ends were placed in slits prepared for them in the rubber stoppers
together with the two rubber tubes containing the potential terminals. Bubber
cement was then placed over the surface and allowed to dry. A strip of rubber
sheet, also covered with rubber cement, was placed so as to surround each rubber
stopper and a portion of the end of the calorimeter, and on being cemented together
formed a sleeve. This rubber sleeve was then firmly wired in place around the rubber
stopper and wires, and also around the calorimeter tube. The ingress for the water

BETWEEN THE FREEZING AND BOILING-POINTS.

207

was by the glass side tube in the inflow end. The egress was by a glass-tube placed
through a bole in the rubber stopper closing the outflow end.

The calorimeter was held in place in the water-jacket by heavy rubber caps specially
made to withstand hot water. On the inflow end of the calorimeter the cap was
placed at the end so as to include the whole length of tube in the water-jacket. On
the outflow end the cap was drawn up to the vacuum-jacket. Side tubes, cemented
into the rubber caps, served to hold them on the calorimeter. The calorimeter was
shoved lengthwise through the water-jacket and the caps sprung into place over
the ends.

The length of outflow tube protruding from the jacket was heavily lagged with
flannel strips wound round it. As the outflowing water was made to flow the
complete length of the outflow tube over the wires leading in the electric current,
and as the tube itself was well protected from outside influences, the loss of heat from
the water in the outflow tube was made as small as possible. This was shown very
effectively by withdrawing the outflow thermometer, when the water was heated
through about 8°, and determining the temperature at different points down
the tube.

The glass-tube placed through the stopper closing the outflow end of the calori¬
meter was connected with a short rubber tube to the coil of tubing in the water
cooler, which in turn was connected in a similar way to the switch-over device.

Water-jacket and Circulating System ivitli Electro-thermal Regulator. — The
water-jacket was an oval tube of g^-inch copjDer, 2 ft. 9 in. long, with two lateral
tubes 1 inch from each end on the under side. The jacket was 6 centims. wide and
8 centims. fligh- On the other narrow end of the oval two other lateral ojjenings
were made, one in the middle for a thermometer to obtain the temperature of the
jacket water, and the other, which could be closed or opened at will, for an exit for
accumulated air from the circulating water. The water in the jacket was circulated
by means of a centrifugal pump run by, a water-motor attached to the high-pressure
mains in the laboratory. The water was drawn from the bottom of a large 10-gallon
copper tank through the jacket to the pump, when it was thrown back again into
the top of the tank. The whole system of circulating tubes formed a chain round
which the water was constantly circulating. No water either left or entered the
system, except that lost by evaporation, and that was exceedingly small except for
the higher points. The circulating tubes were about 4 centims. in diameter, and the
pumping was sufficient to supply a solid column of water from the tubes into the
tank. The tank as well as the water-jacket and circulating tubes were all heavily
lagged. For the higher temperatures a device was fixed to the tank to make up for
the evaporation from the hot water, and to keep it always at the same level. This
was most important, to prevent the exposure of part of the bulb of the thermo¬
regulator by the lowering of the water level in the tank.

In order to maintain the jacket at a constant temperature, a thermo-regulator was

208

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

fitted up in the large tank, similar to the one described by Gouy (‘ Journ. de
Physique,’ vol. 6, p. 479, 1897). Fig. 11 contains a drawing of this regulator with
its attachments. A bulb of glass A, containing about 300 cub. centims. of toluene
resting on a mercury surface B, is connected to a fine heavy-walled tube about
1 millim. inside diameter, through which the mercury at B is made to pass by the
expansion or contraction of the toluene. A three-way glass tap allows the mercury
to pass either into the reservoir C, or up into the tube E. A platinum wire point is
attached to a copper wire and drawn up and down about 4 millims. by a pivot on
the wheel F, worked by a worm-wheel from the pulley C1.

In this method, which is the distinctive feature of the Gouy regulator, the
platinum point never sticks to the mercury surface, and consequently gives a sharper

and more definite electrical connection between the thread of mercury in E and the
wire. Connections were taken from this to a telegraphic relay, which was so
arranged as to throw in and out a heating lamp placed in the tank. The arrange¬
ment is shown in fig. 12.

When the relay is inactive, the terminals of the lamp are short-circuited by the
arm extending between the mercury cups a and b , and the full current is permitted
to pass through the lamp B. When the relay is excited, the arm ah is raised and
the circuit broken at a , so as to bring the lamps A and B in series. Lamp A was a
10 -candle-power lamp of 200 ohms resistance, while B was either a 3 2 -candle-power
lamp of 100 ohms resistance, or a 50-candle-power lamp of about 60 ohms resistance.
Either of the two heating lamps in series with lamp A was reduced in heating power
over one-fourth of its full amount. The relay was made active by the closing of the

BETWEEN THE FREEZING AND BOILING-POINTS,

209

connection between the wire and mercury column, and was supplied by either one or
two accumulators. This form of regulator works exceedingly well, and is much to
be recommended.

The electric heating lamp lias practically no lag, so that the effect known as
“ hunting ” was not apparent. The bulb of the regulator was long and narrow, and
extended the whole height of the water in the tank. For high temperatures, lamp B
could not supply enough heat to make up for the loss from the circulating system to
the surrounding air, so that an auxiliary gas flame was necessary, which was placed
under the centrifugal pump, and was supplied by a large constant-pressure gas
regulator situated in the basement of the building. The final amount of heat
necessary to keep the apparatus at the temperatures of the experiment required,
was supplied by the lamp B, which thus acted as a fine adjustment over and above
the heat supplied by the gas flame. For very high points it was necessary to main¬
tain a second gas flame under the large copper tank, which was arranged on three
brick supports for this purpose.

Fig. 12.

The water-motor which operated the pump in the circulating system ran at a very
constant speed, on account of the very steady pressure of the water in the high-
pressure mains. This aided very much in ensuring perfect regulation. Several
stirrers in the apparatus were also operated by the water-motor from a pulley
directly connected to it. One of the stirrers was placed in the tank and shown
connected with the pulley C, in fig. 11. This helped to keep the water throughout
the tank thoroughly mixed. It was connected to the pulley of the water-motor
by a leather strap. Another cord was taken to the stirrer in the standardized
resistance oil-bath, and a third to the stirrer in the cooler in the outflow end of the
calorimeter shown in fig. 13. A general idea may be obtained of the arrangement
of the apparatus by reference to fig. 13.

The accompanying photographs are added to give some idea of the appearance
and arrangement of the apparatus. Fig. 14 is a side view, and fig. 15, a bird’s-eye
VOL. CXCIX. - A. 2 E

210

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

view, taken from the ceiling. Corresponding parts ol the apparatus are indicated by
the same letters in the two plates.

Fig. 14, is taken looking towards the slate slab along one side of the laboratory on
which the greater part of the apparatus was arranged. The water-motor A, which
supplied the driving power for the circulation and all the stirrers, is seen on the
extreme right. B is the heater, containing a centrifugal pump driven direct from
the motor, which delivers the water into the regulator tank C (fig. 14) through a
large rubber hose. Behind the tank is the hypsometer T, which was employed for
pre-heating the distilled water at the higher temperatures, and the water-bath P,
containing the tube resistances for regulating the flow of the distilled water. The
distilled water reservoirs were on the floor above. D is the ebonite box containing;
the standard resistances for current measurement immersed in oil with a stirrer
driven by the central pulley (fig. 5). E is the copper water-jacket (fig. 11) containing
the calorimeter, swathed in flannel, and connected by rubber hose on one side to the
regulator tank C, and on the other to the circulating pump B. F is the switch-over

Fig. 13.

O

tap (fig. 7) for delivering the flow into either of the two flasks G (fig. 8), and
automatically recording the time of the switching over on the electric chronograph,
the cylinder of which is marked Q in fig. 15. In the background of fig. 14, on
the slate slab to the left are seen, H the Thomson- Varley slide-box, and K the
100,000-olnn galvanometer. Nearer the middle at M is the 20-ohm galvanometer
for the platinum thermometers. These are all beyond the range of fig. 15. L is the
zigzag platinoid rheostat for regulating the main current so as to obtain the desired
rise of temperature in each case.

On the small table at the side in fig. 15 is seen the compensated resistance box Pi
for the differential platinum thermometers, with small auxiliary boxes for current
regulation. S is the rubber tube containing the leads to the heating lamp in the
tank C. The relay, (fig. 14) worked by the regulator in the tank, and the shunt
lamp are seen at Q in fig. 14, but are hidden by the jacket E in fig. 15. The Clark

BETWEEN THE FREEZING AND BOILING-POINTS.

2 e 2

212

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

cell tanks and regulator are not shown in either view. They were at the other end
of the laboratory.

Water Supply for Calorimeter. — The water supplied to the calorimeter passed
through 40 feet of pure block-tin tube, \ inch in diameter, coiled up in the constant
temperature tank. After passing through this and taking the temperature of the
jacket water, it was passed from the tank to the water-jacket and into the
calorimeter through a rubber tube placed inside the water circulating tube. By
this means the water, after entering the tank, was entirely in the circulating system
until it flowed out of the calorimeter. The head of water to maintain a steady flow
was supplied from a reservoir placed on the floor above, and was connected to the
apparatus by a glass-tube passed through a hole in the floor. In order to vary the
supply, the water was passed through a series of fine tubes acting as resistances,
which could be short-circuited by larger tubes connected across them. These larger
tubes, offering no resistance, were thick-walled rubber tubes and connected to the
ends of the fine tubes by T -pieces. When the water was to be passed through the
fine tubes, the rubber tube was simply closed with a pinch-cock. The resistance
tubes were two principal ones, 1 metre long and 1 millim. in section, and three lesser
ones for fine adjustments. These were all immersed in a water-bath to keep them
from changing in temperature suddenly, and thereby producing a change in the flow
by changing the viscosity of the water.

No device was used to maintain a constant head, as a slight falling-off in the flow
was rather an advantage than otherwise, as it tended to compensate for the slight
falling-off in the electric current supplied to the calorimeter by the large accumulators.
Two large bottles, holding about 4 gallons each, formed the head and were connected
in parallel. A layer of heavy paraffin oil was put over the water to prevent
absorption of air by the distilled water, which was always used for the experiments.
This was supplied to the bottles in sufficient quantity for about two experiments, and
was run in under the oil through a T-connection in the tube connecting the two
bottles together and with the supply tube for the apparatus. The water was first
boiled in a large copper tank, and while still boiling was siphoned over into the
bottles. It was allowed to cool before being used for an experiment. This method
of boiling the water was used for all the earlier experiments below 60° C. ; but it was
found impossible to obtain steady conditions of flow above this limit, owing to the
liberation of air inside the calorimeter even from the boiled water. This was some¬
what surprising and delayed the attainment of the final measurements at the high
temperatures. • It appears that to remove the last trace of air from boiled water, it is
necessary to submit it to extreme agitation. Sufficient agitation was supplied to the
water, as it ran through the fine-bore tube of the calorimeter, to set free some of the
air retained by the boiling water when it was run into the bottles under the oil.

It was found necessary to devise some method of preparing absolutely air-free
water before readings at the higher points could be obtained, To do this I found.

BETWEEN THE FREEZING AND BOILING-POINTS,

213

214

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

after repeated trials with other arrangements, that it was necessary, in order to
separate the air from the water, to drive the latter into steam and condense it again
in an air-free medium, and at the same time draw off the air before it had a chance
to dissolve again. The method adopted finally was to exhaust the air over the oil in
the two bottles by connecting with a water-pump while the boiling water from the
copper tank was running in. In this way, by introducing a slight constriction in the
siphon tube, such as a connection made from a small-bore rubber tube, the water in
the tube separated into steam just past the constriction, and was condensed in the
cold water already in the bottles from a previous filling. The air was at the same
time sucked up through the oil and carried off by the pump. It was a matter of
considerable surprise to see the amount of air thus drawn off.

After adopting this method of filling the bottles, there was no further trouble from
air appearing inside the calorimeter, even as high as 90° C. When working at high
temperatures, the distilled water, before it passed into the constant temperature
tank, was passed through a spiral of tin tube in a steam-jacket. Instead of increasing
the heat-loss of the circulating system by introducing cold water into the tank, a
small quantity of heat was supplied by this means to the tank by the water flowing-
in from the steam-jacket. Moreover, the steam-jacket was a check on the state of
the water, air bubbles being generated if the water was not perfectly air-free. The
water which was run through the calorimeter was never used a second time, although
it might just as well have been. Fresh distilled water was so easily prepared by
means of a small water still in the laboratory that it was deemed unnecessary to use
it twice.

A constant-level head was arranged near the apparatus to supply water for the
cooler for the outflowing water ; at the same time it also supplied the water circula¬
tion for the standardized resistance oil-bath, for the constant-level device of the tank,
and for the condenser on the steam-jacket used at the higher temperatures.

Method of Making an Experiment. — Obviously various expedients were necessary
in order to have the jacket maintained at a constant temperature at any point on the
scale between 0° and 100° C. For the experiments near 0° C. the regulator was
removed entirely from the tank, which was then filled with cracked ice and water.
A wire sieve was placed over the stirrer in the bottom of the tank, so as not to have
its action interfered with by the ice, as well as to prevent any ice from being drawn
into the circulating tubes. Wonderfully steady conditions were produced in this
way, and maintained without the variation of a hundredth of a degree for over an
hour at a time. Between the measurements with each flow of water used, which
lasted about an hour, the tank was replenished with ice. Not a great deal of ice was
required for this replenishing, since only about one-quarter of the tank-full of ice was
melted in the hour. From 50 to GO lbs. of ice were melted during a complete
experiment, although considerably more was used to cool the apparatus down to the
ice-point before the experiment was started. From 100 to 150 lbs. of ice was

BETWEEN THE FREEZING AND BOILING-POINTS.

215

generally required for a complete ice determination. It may be said, however, that
these experiments were among the easiest to obtain.

At temperatures intermediate between the ice-point and room temperature, the
regulator was used, and a stream of cold water run through a coil of metal tube
placed in the tank. Water direct from the water mains usually sufficed for the
cooler, but for lower temperatures the tap water was run first through a coil of tube
immersed in a tank of ice. The desired temperature was attained through the
equilibrium between the heat supplied by the lamp in the tank and the heat absorbed
by the water flowing through the circulating tube. The tap Avater in the high-
pressure mains varied, from about 5° in winter to about 18° in summer, in the
laboratory where the apparatus was located. A temperature at least 2° lower than
the desired temperature of regulation was required in the cooling water. The
temperature of the laboratory varied from about 18° to 25° during the different
seasons; but, in general, was steady to 1°, and often less during an experiment
lasting 3 or 4 hours. When not using cooling water in the tank, the temperature of
the apparatus had to be maintained a few degrees above the laboratory, so that the
regulator could make up for the loss of heat to the surrounding air. Hence a
temperature of about 26° or 27° in the tank was the most convenient point at which
to work, which for an 8° rise in the calorimeter gave a mean temperature of about
30° C.

As the temperature of the jacket was increased more and more above the room
temperature, more and more gas was required to aid the electric-heating lamp in
maintaining a constant temperature. At a temperature of the jacket of about
90° C. two gas flames were required, and, in addition, the 50 c.-p. lamp. Even for
the highest points the regulation proved to be most satisfactory, so long as the gas
pressure did not vary. It was found that the high temperature experiments had to
be taken at times when there was no other gas being used in the building, as even in
using the large constant-pressure gas governor fluctuations in the gas pressure in the
building caused perceptible alterations to the regulator. Where the fluctuations
were small and regular, the jacket water seldom varied as much as '01° during the
time for obtaining the observations for each flow, even at the highest point
attained. The wonderful efficiency of the regulator and circulating system, together
with the preparation of air-free water, made it possible to obtain the observations at
the higher temperatures with almost the same degree of accuracy as at the lower
points.

At whatever temperature the jacket was maintained, and before the electric-heating
current was turned on, the calorimeter water was allowed to flow through the
apparatus for some time after the steady conditions were attained by the regulator.
The balance-point so obtained for the differential thermometers we have already
termed the “ cold” readings, in contra-distinction to the readings obtained when the
electric-heating current is turned on. The second balance-point, together with the

216

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

requisite number of bridge resistance-coils required to compensate the rise in tem¬
perature in the water, we will term the “ hot ” reading. The conduction errors at the
ends of the apparatus remained the same for both readings, except (as has already
been pointed out) the conduction error due to the rise in temperature, d9, in the out¬
flow end. It was therefore possible for any temperature over the scale between
0° and 100° to eliminate the conduction errors at the ends of the apparatus by the
“ cold " readings in a, very simple manner, and reduce the only conduction error to be
considered, i.e., that due to the rise (16, to the same amount all over the scale. Unless
this procedure had been followed, large errors would have been incurred, especially at
the inflow end where the conduction effect was the largest.

The adjustment of the electric current to any given flow was made by varying the
number of accumulators, or by inserting a number of platinoid strips, ’02 ohm, in
series in the rheostat. While the conditions became steady in the calorimeter after
turning on the electric current, requiring 10 to 15 minutes, the weights of the empty
flasks were obtained as already described. One of these was then affixed to one of
the nozzles on the calorimeter. The temperature of the cooler was then adjusted, and
the preliminary readings of the potential difference across the standard resistance and
calorimeter obtained. The chronograph being started, the accurate balance of the
two Clark cells in series was obtained, together with the temperature of the Clark-cell
bath. The other conditions remaining steady (including the thermo-regulator in the
tank, the jacket circulation and the different water circulations to the resistance oil-
bath, cooler, constant-level device and steam-jacket when used, as well as the thermo¬
regulation in the Clark-cell bath) the flow was switched over into the weighed flask
at a given moment, which was recorded automatically on the chronograph. The
complete set of observations then followed in order for 15 minutes. These, besides
the temperature of the cooler at the beginning and end of the interval, were made
every minute — first minute the deflection of the galvanometer at the nearest millimetre
on the bridge-wire for the balance-point of the differential thermometers, then, in
succession at every even minute, the potential balance of the standard resistance, the
reading of the differential thermometers, potential across the calorimeter, reading of
the thermometers, potential across standard resistance, and so on, including at the
half-minutes the reading of temperature of standard-resistance oil-bath, jacket water,
air temperature and temperature of thermometer resistance box, although this last
was not really necessary.

At the end of the 15 minutes, to the nearest second by the watch used in starting
the interval, the flow was switched over to the other nozzle, and the time automatically
recorded. The balance-point of the Clark cells was then obtained, together with their
temperature. On changing the full flask for a second empty one, a second set of
readings for 1 5 minutes was obtained, without otherwise altering the conditions. The
extreme steadiness of the potential balance for the Clark cells made it quite unnecessary
to have it recorded oftener than just before and just after the 15-minute interval.

BETWEEN THE FREEZING AND BOILING-POINTS.

217

For each 15 minutes 8 readings in all were obtained of the differential thermometers,
which were separated by 2 minutes each, the last one being taken just before switching
over the flow, 3 readings of the potential across the calorimeter separated by 4 minutes,
and 4 of the potential across the standard resistance also separated by 4 minutes.
These readings were always corrected to the middle of the interval to correspond to
the time of average flow. The thermometer readings sometimes increased and some¬
times decreased during the interval, depending on the flow, but the change was never
more than ‘02° or ’03° over the 1 5 minutes, and generally much less. The question
of the lag of the thermometers and of the thermal capacity of the calorimeter, which
amount to such large corrections in all older calorimetric methods, was reduced there¬
fore to a negligible quantity. The potential readings on the Thomson-Varley slide-
box of the calorimeter and standard resistance generally fell off in a regular way
throughout the interval, although quite often they remained steady altogether. The
fall was seldom as much as 1 part in 1000 during the entire interval; hence a greater
number of readings was unnecessary, since all the readings were taken at stated times,
which were even minutes recorded from the seconds’ hand of the watch used to start
the flow over the interval.

In the earlier experiments, where two observers took the observations, simultaneous
readings could be made of the temperature and potential difference every minute.
This arrangement was of course preferable to the other, but where the conditions
remained steady a large number of readings was really not necessary. When any
sudden change in the conditions occurred, such as in the regulation, electric current
or flow, so as to make the readings unsteady, these readings were either repeated
during another interval where it was possible to restore the conditions in a short time,
or they were abandoned altogether until such other time as the complete experiment
could be repeated. My aim was, as far as possible, to produce a series of measurements,
over the entire range of temperature between 0° and 100°, under as steady and
uniform conditions as possible.

The number of flows usually taken in a complete experiment was two, but some¬
times three. Other flows were tried to test the theory of the method beyond the
limits of flow chosen for the actual experiments. In all the flows two intervals of
15 minutes were taken, which, when worked out, gave a complete check on the
steadiness of the conditions and the accuracy of the observations.

None of these measurements depend very much on the absolute readings of mercury
thermometers except the Clark cell. The temperature of the standard-resistance was
always taken with the thermometer used in its calibration with the standard ohm,
which in turn had such a small temperature coefficient that it made it of very little
importance whether the thermometer used to obtain its temperature was in error by
as much as ‘5°. As a matter of fact, by direct comparison, all the thermometers used
agreed with our standard thermometer to within T°. The thermometer used to
obtain the temperature of the jacket-water between 0° and 50° C., was a new Muller

VOL. OXCIX. - A. 2 F

218

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

graduated to ‘1° and reading to Youths. Its error was determined by comparison
with the standard thermometer, and its readings were found to be "03° too large. The
standard thermometer was a Geissler reading to Tooth between — 2° and 50° C.,
which has been in the possession of the laboratory for several years, as before stated.
In 1896 both Professor C allend ar and I separately determined its error, and reduced
its readings to the nitrogen scale over the entire range by comparison with a platinum
thermometer. The error was very consistent, and showed its readings too high by
•11° to ‘12° from 0° to 50°. This thermometer was used in the Clark-cell bath and
lias already been referred to in that section. For jacket-temperatures between
50° and 90° C., a second Muller thermometer, reading between 50° and 150°, was
used, which was graduated to -g-th of a degree.

Specimen Tables of Observations.

\_Added April '20th, 1901. — H.T.B. Owing to the necessity of condensing the
tables since the communication of this paper, I have considered that it would be of
advantage to give specimen tables of observations as recorded during a complete
experiment. I have therefore included here two sets, made as typical as possible,
which illustrate more clearly the remarkable steadiness of the conditions. The
experiments selected are those given under date October 27, 1899, and March 10,
1900. They include two calorimeters and different-sized flow-tubes (Calorimeters C
and E), as well as flat and round heating wires. One of the sets includes the
observations taken for the measurement at the highest point of the range. In the
first set box 1 was used, on the second set box 2. The order in which the readings
were obtained has been described in Section 4. In all cases the time of taking the
readings was as closely as possible on the even minute. The time for the start and
finish of an interval, during which the flow was measured, is of course given, as
recorded on the chronograph, to ’01 second. The reading of the contact point on the
bridge-wire (b.-w.) is given in centims. (10 centims. = 1CC.), and the deflections of
the galvanometer noted, it being accurately set to the nearest millimetre by means
of the vernier reading to hundredths of a millimetre. The sensitiveness of the
galvanometer remained very constant, but was repeatedly checked during a set of
observations. The balance-point on the bridge-wire was calculated by interpolation
from the observations of the deflection. In the electrical readings of the potential,
S stands for the balance-point for the difference of potential across the standardized
resistance, and P for the same across the calorimeter heating-wire. These readings
are of course uncorrected for the errors in the Thomson- Varley slide coils. The
temperature of S is that of the oil in which the standardized resistance was immersed,
and the temperature of the Clark cells is here uncorrected for the thermometer error.
The inflow temperature is the same as the temperature of the jacket. In some cases

BETWEEN THE FREEZING AND BOILING-POINTS.

219

the air temperature was recorded just at the calorimeter, as well as the temperature
of the balance case, but in general these never differed much from the temperature cf
the air as recorded on the barometer case, which was always taken with the reading
of the barometer. The weights are given here in grammes, just as recorded during
the experiment. The Abridged Tables sent in with the paper give the summary of
these observations, together with the necessary corrections.]

\Added May 2S th, 1902. — H.T.B. The comparative failure of my attempts to
obtain consistent results in the experiments carried out previous to my discovery of
the effect of stream-line motion on the distribution of heat in the fine bore flow-tube,
appears to me to have been largely due to the fact that the stranded conductor was
in nearly all cases annealed before being inserted in the calorimeter. This caused
the different strands to lie together more in the nature of a solid conductor. It is
probable that better results would have been obtained in these early experiments
had the wires been stiffer, the flow-tube smaller, and had it been possible to
distribute the strands more thoroughly in the water column, and at the same time to
prevent them from changing their relative position in the tube between the
experiments.]

2 F 2

220

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

Specimen Table of Observations XXXYI.

Flat Heating Wire.

October 27, 1899. Calorimeter E (3 millims.).
Mean Temperature, 290-92.

Large flow.

Small flow.

Cold readings.

1

W

eights.

Cold readings.

Weights.

Time.

B.-W. Deflea.

First Second

Time.

B.-W. Deflec.

■

First Second

interval, interval.

interval, interval.

2-33

23 600 11 c.p.

Flask (1) . .

99-379 —

355

23"600 8 c.p.

Flask (1) . .

99 162 —

34

11 e.p.

„ (-') • •

120-793

56

„ 8 c.p.

„ (2) • •

120-255

35

23-700 25 s.a.

„ (1) Ml

659 755 —

57

„ 8 c.p.

„ (1) full

444-392 —

33

,, 24 s.a.

„ (2) „

6SO-500

58

23 700 29 s.a.

„ (2) „

— 464-695 !

37

,, 24 s.a.

59

23 600 8 c.p.

33

23 600 12 c.p.

4-00

„ 8 c.p.

3J

,, 10 c.p.

Barometer . . 76"57

01

>> 8 c.p.

Barometer . . 7657

40

23'700 25 s.a.

Temperature . 1S'2

Temporal

ure . 18’2

41

„ 25 s.a.

42

,, 26 s.a.

43

23-600 11 c.p.

Temperature balance case not taken.

Temperature balance ease not taken.

4'4 Electric heating current on.

4-02

Electric heating current on.

3'04'00'42 Start. Pot. bal. C.C. 63004. Bos coil 80.

4T9'00'62 Start. Pot. bal. C.C. 62998. Bos coil 80.

05 220 00 lls.a.

20

22-000 2 s.a.

06

Pot. bal. S. 70251

Temp. C.C. . 16T4

21

Pot. bal. S. 55596

Temp. C.C. . 46T4

07

,, 15 s.a.

,, inflow 25'74

22

„ 3 c.p.

,, inflow 25"75

OS

P. 76282

23

P. 60235

09

,, 19 s.a.

,, air . 13-8

24

„ 4 c.p.

,, air . 18"8

io-

„ S. 70246

25

, S. 55591

11

,, 19 s.a.

„ S. . . 14-0

26

,, 6 c.p.

,, S. . . 13*5

12

„ P. 76277

27

„ P. 60231

13

„ 21 s.a.

28

5 c.p.

14

S. 70241

29

,, S. 55589

15

„ 21 s.a.

30

„ 4 e.p.

16

„ P. 76274

31

P. 60227

17

„ 23 s.a.

.. S. . . 14-0

32

„ 4 c.p.

,, S. . . 13 5

18

S. 70238

33

. S. 55586

18-30

,, 30 s.a.

33-30 „ 3 c.p.

3T9"00T2 Flow switched

over.

4-34 00-12 Flow switched

over.

3-2200-

29 Start. Pot. bal. C.C. 63004. Second interval.

4-59'01"05 Start. Pot. bal. C.C. 62997. Second interval.

23 22000 Bal.

5-00

21-900 4 c.p.

24

Pot. bal. S. 70230

Temp. C.C. . 16T3

01

Pot.

bill. S. 55571

Temp. C.C. . 1614

25

„ 1 s.a.

inflow 25'75

02

„ Bal.

„ inflow 25 75

26

P. 76263

03

P. 60211

27

5 s.a.

,, air . 19'0

04

„ 1 c.p.

,, air . 19"3

28

S. 70227

05

S. 55570

29

,. 8 s.a.

„ S. . . 14-0

06

„ 2 s.a.

S. . . 13-5

30

,

P. 76262

07

P. 6020S

31

„ 9 s.a.

08

„ 2 s.a.

32

S. 70224

09

S. 55567

33

14 s.a.

10

„ 5 s.a.

34

>

P. 76257

11

P. 60206

35

,, 22 s.a.

,. S. . . 14 0

12

„ 9 s.a.

36

S. 70219

13

S. 55567

3630

„ 29 s a.

13 30 „ 13 s.a.

3'37 00T6 Flow switched over.

5'14 00'59 Flow switched over.

Pot.

bal. C.C. 63003

Temp. C.C. , 1613

Pot. bal. C.C. 62996

Temp. C.C. . 1613

1

BETWEEN THE FREEZING AND BOILING-POINTS.

221

Specimen Table of Observations LXII. March 10, 1900. Calorimeter C.
Solid Heating Wire. Mean Temperature 9 1 °*5 5.

Large flow.

Small flow.

Cold readings.

Weights.

Cold readings.

Weights.

Time.

B.-W. Deflec.

First

Second

Time.

B.-W. Deflec.

First Second

interval. interval.

interval, interval.

3-OS

22 600 2 c.p.

Flask (1) . .

97-683

5-23

22-400 10 c.p.

Flask (1) . .

97538 —

09

,, 7 c.p.

„ (2) . .

—

121-207

24

„ 12 s.a.

„ (2) •

121-031

10

» 7 c.p.

(1) full

678004 —

25

,, 44 s.a.

„ (1) full

459-847 —

11

22-700 29 s.a.

, (2) „

—

699-58 4

26

,, 40 s.a.

„ (2) „

— 481-545

12

„ 28 s.a.

27

„ 38 s.a.

13

22-600 7 c.p.

28

22-300 5 c.p.

Barometer . .

75-77

29

,, 3 s.a.

Temperature

22-0

30

,, 9 s.a.

31

,, 6 s.a.

Barometei

. . 7577

32

„ 5 c.p.

Temperature . 22'0

Temperature of balance case taken as

33

„ 2 c.p.

that of air.

34

Bal.

35

1 s.a.

36

,, 4 s.a.

37

„ 5 c.p.

3T4

Electric heating current on.

5-38

Electric heating current on.

3-38 00-30 Start. Pot. bal. C.C. 63562. Box coil 80.

6'09'00 45 Start. Pot. bal. C.C. 63557. Box coil 80.

39

29-200 Bal.

10

28'600 16 c.p.

40

Pot. bal. S. 74004

Temp. C.C. . 15‘27

11

Pot. bal. S. 59290

Temp. C.C. . 15-21

41

„ 20 s.a.

inflow 87'42

12

10 s.a.

,, inflow^ 87’43

42

P. 76674

13

, P. 61350

43

,, 8 s.a.

S. . . 7-0

14

,, 12 s.a.

„ S. . . 6-6

44

, S. 73990

15

, S. 59287

45

„ 9 s.a. ■

16

„ 29 s.a.

46

, P. 76658

17

„ P. 61347

47

„ 20 s.a.

18

28'500 11 s.a.

48

„ S. 73976

19

, S. 59283

49

„ Bal.

20

,, 13 s.a.

50

, P. 76647

21

,, P. 61343

51

„ 24 s.a.

S. . . 7-0

22

28-400 15 s.a.

» S. . . 6-6

52

, S. 73965

23

, S. 59281

52'30 „ 25 s.a.

23 30 „ 46 s.a.

3-53-00-69 Flow switched over.

6"24'00-40 Flow switched

over.

3'56 00'82 Start. Pot. bal. C.C. 63562. Second interval.

6-27-00-42 Start. Pot. bal. C.C. 63557. Second interval.

57

29-100 3 s.a.

28

28-200 7 c.p.

58

Pot. bal. S. 73949

Temp. C.C. . —

29

Pot

bal. S. 59277

Temp. C.C. . 15"22

59

,, 10 s.a.

inflow 87’41

30

28-100 14 c.p.

,, inflow" 87"43

4-00

„ P. 76621

31

, P. 61338

01

,, 21 s.a.

S. . . 7-0

32

„ 14 c.p.

„ S. . . 6-6

02

, S. 73939

33

, S. 59274

03

29 000 10 c.p.

34

„ 21 c.p.

04

„ P. 76613

35

, P. 61336

05

„ 1 s.a.

36

„ 11 c.p.

06

, S. 73930

37

, S. 59272

07

„ 8 c.p.

38

„ 6 c.p.

08

P. 76605

39

P. 61334

09

Bal.

S. . . 7-0

40

,, 10 s.a.

,, S. . 6'6

10

„ S. 73921

41

S. 59270

10"30 ,, 5 s.a.

41-2

0 ,, 22 s.a.

4'11'01 30 Flow switched

over.

6-42-01T2 Flow switched over.

Pot. bal. C.C. 63561

Temp.

C- C. . 15-21

Pot. bal. C.C. 63557

Temp. C.C. . 15 20

222

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

Abridged Tables Showing Method of Correction,

To illustrate the method of correction and calculation of results, we here append
the corresponding Abridged Tables giving the correction and reduction of the means
of the observations taken during each interval for the two specimen tables of
observations already given. As it would have been impracticable to place on record
the complete observations for the whole work, the abridged tables only were sent
in with the paper. The greater part of these tables consisting of small corrections
of no intrinsic interest, it was felt to be unnecessary to reproduce them in full.
They have been preserved in the Archives so as to be available for reference and
verification if required. The samples here given will sufficiently illustrate the
nature of the information they contain.

Abridged Table XXXVT., Series 5. Inflow Temperature, 25°'75. Mean Temperature, 290,92. October 27th, 1899

Calorimeter E.

BETWEEN THE FREEZING AND BOILING-POINTS. 223

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Abridged Table LIII., Series 8. Inflow Temperature, 87 ’*42. Mean Temperature, 91L,55. March 10, 1900.

Calorimeter C.

224 DR, H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

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BETWEEN THE FREEZING AND BOILING-POINTS. 225

Sec. 5. — Experimental Proof of the Theory of the Method.

In Section 2 it was shown that the conditions to be studied in the general
difference equation of the method were the relations of the heat-loss to the rise of
temperature and to the flow. In the present section I wish to summarize the
different experiments which have a more particular bearing on the theory of the
method. In the first place in regard to the question, which arises in all experiments
where a quantity of water is heated by an electric current conveyed in a wire, of the
excess temperature of the wire over the water, it may be said that in the present
method the measurement of the electrical energy is completely independent of any
value to be assumed for the resistance of the heating-wire, and not only that, but
owing to the steady temperature conditions inside the apparatus, no uncertainty of a
change in resistance in the wire with a change in temperature is introduced. When
the temperature inside the calorimeter has arrived at a steady state, only such
energy is used in warming the water as is supplied to the calorimeter by the electric¬
heating current. The fact that the results were completely independent of the
resistance of the heating-wire was shown by using heat-wires of very different
resistances.

In regard to the insulation of the platinum heating-wire and of polarization and
similar effects, it was considered that these played no part in the results. The
resistance of the water column through which the heating-wire passed was
enormously high and equal to a column of water 50 centims. long and 2 millims. in
diameter, hence in comparison with the resistance of the central heating-wire, which
varied from '4 to '8 ohm, was quite negligible. This is true even if it is admitted
that the conveyance of the electric current by the water itself could have produced
any error on the final result.

Polarization by the naked wire in the water, I am satisfied, did not take place.
Not the slightest trace of gas was ever generated in the calorimeter which could not
be referred to the liberation of air in the water, and this was verified by watching
the column of water in the fine tube when the calorimeter was removed from the
water-jacket and a large electric current passed through. The effect of reversing the
electric current in the apparatus, and making it flow either with or against the water
flow, was tried in some of the earlier experiments, but it was found to produce no
effect on the heat-loss as measured by the difference between the electrical and
thermal measurements. The effect, if any, on the electrical readings was entirely
negligible.

The first experiments which were tried to test the method, were on the relation of
the heat-loss to the flow, and were made by varying the flow over a wide range and
at the same time keeping the inflow temperature and rise of temperature constant.

The following list of calorimeters, used in the present work, will aid in describing
these and subsequent experiments : —

VOL. CXCIX. — A. 2 G

226

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

Calorimeter A. — The first one of the set of three obtained from Germany, with
the fine-bore tube 2 millims. in diameter. This was the first one tried in
the trial experiments in 1897.

Calorimeter B. — Second of the same set, with same dimensions, and used in the
preliminary measurements of the mechanical equivalent.

Calorimeter C. — This was the first one made by Eimer and Amend, after the
later design. It had a 2 millims. -bore tube, and has been used for the
greater number of the later experiments.

Calorimeter D. — The second E and A calorimeter, with the fine-bore tube 2'8
millims inside diameter and P305 in the vacuum chamber.

Calorimeter E. — The third from E and A, with the fine-bore tube 3 millims.
inside diameter.

The results of the experiments made to determine the relation of the heat-loss to
the flow, are now given. They have been taken from the experiments detailed in
Tables I., II., III., and IV. in Section 8. The results are taken from the observa¬
tions on two calorimeters with fine-bore flow-tubes of different sizes, the heating wire
in the two cases being made up of either the six strands of T5-millim. platinum wire
or the solid wire, and held central by the silk-covered rubber elastic wound round it.
The water was therefore completely stirred in its passage through both the tubes,
and stream-line motion avoided.

For convenience in showing the relationship I have expressed the difference
between the electrical and heat watts by using the value of J obtained from the
experiment for two flows in place of 4 "2 joules. In this case the value of the heat-
loss per degree rise should come constant for all of the flows as long as the conduction
effect is negligible.

The first set comprises observations made with Calorimeter D at a mean
temperature of 28° C. The flow was varied from '67 gramme per second to '25, and
the rise of temperature was kept approximately the same by adjusting the electric
heating current.

Calorimeter D. — Mean Temperature, 28°'01 C. February 15, 1899.

8 = — -00485.

J = 4-1797.

d6.

Q.

(EC - JQ d6)/cW.

Difference from
mean, ‘07128.

7*5234

•674106

•07122

- -00006

7-8882

•496655

•07147

+ -00019

7-7745

•399290

•07130

+ -00002

7-9463

•390196

.07113

- -00015

8 0033

•248234

•07197

+ -00069

BETWEEN THE FREEZING AND BOILING-POINTS.

227

It will be seen that over the range of flow from * 67 gramme to ‘39, the value of
the heat-loss remains constant to within the limits of error of the different measure¬
ments, and gives a mean value of '07128 in watt-seconds per degree rise. The
largest variation from the mean is for a flow of '49 gramme per second, and amounts
to '00019 watt on a total supply of 2T watts per degree, which is less than 1 in
10,000. For the small flow, the difference from the mean of the other flows amounts
to '00069 watt on I T watts. This shows an increase in the heat-loss of nearly
7 parts in 10,000, and is much too large to be included within the limits of error. It
is evident, then, that for flows below '3 gramme per second, the conduction effect
commences to be measurable, and cannot be eliminated by the method of “ cold ”
readings.

For Calorimeter C, the measurements for the different flows are given under date
February 20, 1899. These are for double intervals of 15 minutes each, and include
the same limits of flow as for Calorimeter I). I have taken S = '00490, instead of
— '00469 as given from these measurements, so as to give results comjmrable with
the other sets of observations made at about this time.

Calorimeter C. — Mean Temperature, 29o-09 C. February 20, 1899.

§ = '00490. J = 4T794.

(16.

f

Q.

(EC - JQ d6)/dd.

Difference from
mean, '04997.

8-2608

•398498

•04972

- -00025

8-2560

•398540

•04988

- -00009

8-2199

•501957

•05016

+ -00019

8-2301

•501026

•04998

+ -00001

7-9646

•666042

■ 05009

+ -00012

7-9775

■664388

•04999

+ -00002

8-2281

•258114

•05057

+ -00060

8-2284

•257947

•05070

+ -00073

The largest variation from the mean value of the heat-loss is -—'00025 watt, and
amounts to a little more than 1 in i 0,000. For the small flow, the divergence from
the mean of the other flows amount to '00067 watt, and shows that the heat-loss has
been increased, which agrees very closely with the result obtained for Calorimeter D.
The agreement of the results for the two calorimeters, the one with a 3-millim. bore
tube and the other with a 2-millim., seems to prove fairly conclusively that the
increase in the heat-loss taking place below a certain limit of flow, cannot be
attributed to a change in the radiation loss from the fine bore tube, but can only be

2 g 2

228

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

referred to conduction from the outflow tube, which was the same size in both
calorimeters.

The observations under February 22 and March 2 were taken with Calorimeter C,
using the limits of flow which I have since adopted for the present measurements.

Calorimeter C. — Mean Temperature, 29°-ll C. February 22, 1899.

8 = -00490.

J = 4-1794.

eld.

Q-

(EC - JQ d0)/dd.

Difference from
mean, '04938.

8-2680

•392606

•04939

+ -ooooi

8-2635

■392663

•04937

- - ooooi

8-1938

•496708

•04928

- -oooio

8-1844

•496591

•04946

+ -00008

7-9031

•660865

•04945

+ -00007

7-9083

•658741

•04932

- -00006

Calorimeter

C. — Mean Temperature, 290,21 C.

March 2, 1899.

8 = — -00499.

J = 4-1790.

dd.

Q.

(EC - JQ dd)/dd.

Difference from
mean, -04968.

8-4310

•375154

•04963

- -00005

8-4304

•375076

•04967

- -ooooi

8-3979

•472489

•04967

- -ooooi

8-4060

•471670

•04986

+ -00018

8-3390

•590477

•04960

- -00008

8-3439

•589356

•04969

+ -ooooi

The agreement of the heat-loss is very satisfactory, and the variations from the
mean value are easily within the limits of error of all the measurements, and are all
less than 1 part in 10,000.

The three sets of readings for Calorimeter C show a small difference between the
values of the mean heat-loss. This shows, as has been pointed out, that the absolute
value of the radiation loss for one calorimeter cannot be relied on from time to time,
but will vary, for many reasons. Ffowever, this never produced any error in the
measurements of the specific heat of the water, on account of the method adopted of
always eliminating the heat-loss from at least two different flows taken within a

BETWEEN THE FREEZING AND BOILING-POINTS.

229

short time of each other, between which the temperature of the calorimeter never
varied.

In selecting the limits of flow to be used in all the measurements, the accuracy of
the limits was tested for two other temperatures by recording observations for three
different flows at 13° and at 60° C. In these cases the theory of the method was
o-iven a severe test.

o

I have tabulated here the two sets, one for a mean temperature of 13° C. and the
other for a temperature of 60° C. I have taken the values of 8 for each set from the
variation curve.

Calorimeter C. — Mean Temperature, 13°-79 C. March 9, 1899.
8 = — -00208. J = 4-1913.

Temperature of surrounding air, 19° C.

dd.

Q-

(EC -4-2 Q dd)/d9.

Difference from
mean, -03940.

8-5768

•372746

•03946

4- -00006

8-5803

•372262

•03947

- -00007

8-5586

•459149

•03923

- -00017

8-5683

•458194

•03930

- -oooio

8-5499

•573318

•03953

+ -00013

8-5616

•571920

•03940

•ooooo

Calorimeter C. — Mean Temperature, 59°"80 C. June 17, 1899.
8 = — -00360. J = 4-1849.

Temperature of surrounding air, 22° C.

dd.

Q.

(EC - JQ dff)jd6.

Difference from
mean, -07254.

8-3805

•612400

•07277

+ -00023

8 • 3835

•611227

•07263

+ -00009

8-3158

•462971

•07236

- -00018

8-3395

•461364

•07240

- -00014

8-3534

•388491

•07261

+ -00007

8-3674

•387534

•07242

- -00012

The variation from the mean value in both sets is less than 1 part in 10,000. It
is a matter of interest to compare the balance points on the bridge-wire for the

230

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

“cold” readings of the differential thermometers, when in place in the calorimeter,
for the different flows at different parts of the scale.

The three “cold” readings at 60° are, for the largest flow 23"681, for the next
23‘582, and for the lowest 23"544. A decrease in the bridge-wire reading means
that the inflow thermometer is at a lower temperature than the outflow. The heat
conduction at the inflow end, through the copper wires leading in the electric current,
which depends only on the difference existing between the temperature of the
calorimeter water and the temperature of the laboratory, can affect the temperature
of the inflowing water less for the largest flow than for the smallest flow, and
therefore explains the difference in the “ cold” readings for the different flows. For
the flows used in the present work, the effect of the conduction at the outflow end on
the temperature, as indicated by the outflow thermometer, must have been very
small indeed, even at the highest points of the range. All of the “ cold” readings at
the high points are slightly less than the interpolated reading from the differential
ice and steam-points. For the “cold” readings at the ice-point, the effect was, as
might be expected, reversed, and conduction of heat into the calorimeter from the
laboratory took place.

The readings given in Table XLIV., on November 18, 1899, are —

For a flow of ‘59 gramme .... 23 ’330

„ „ -39 „ .... 23-392

on November 22, 1899, Table XLY. —

For a flow of -62 gramme .... 23 -329

„ „ -37 „ .... 23-398

The peculiar exception to this seems to be in the experiment made at the ice-point
on March 24, 1899 (Table XV.), when the “cold” reading for the small flow was
lower than for the high. I have not an exact record of the conditions under which
this experiment was taken, but it is possible that the outflow end may not have
been properly lagged, which would produce the effect indicated by the readings. As
the observations, when reduced, give such a consistent measurement of the specific
heat, even in the face of this apparent exception, additional evidence is given of the
necessity of the “ cold ” readings to render the results independent of extraneous
conditions. The “cold” readings for experiments with the jacket water at the
temperature of the laboratory were, as a rule, all at the same point on the bridge-
wire, and identical with the interpolated reading from the differential ice and steam-
points.

The radiation loss increases only very slightly as the temperatures of the calorimeter
and jacket are raised, and this is of course on account of the temperature of the
calorimeter flow-tube being always the same amount above the jacket water at all

BETWEEN THE FREEZING AND BOILING-POINTS.

231

points of the scale (when the water is heated by the electric current). The increase
is from the temperature coefficient of the radiation, and appears to be almost exactly
linear over the range 0° to 100°.

To show the relation between the heat-loss and the rise of temperature, I have
summarized the observations in Tables XXXIII., XXXIV., XXXV., for rises of 8°, 5C,
and 2° respectively. These were made with Calorimeter C, but the flat heating-wire
was used in place of the central heating conductor with rubber elastic.

Relation of heat-loss to rise of temperature.

Calorimeter C. — Mean Temperature, 28=,G C. October 14, 18, and 19, 1899.

dd.

Q.

(EC-4-2Q d0)ldd.

Diflerence from mean
value, '04535.

Larg

3 How.

8-3069

•626436

•04521

- -00014

8-3212

•625128

•04533

- -00002

5-1009

• 636545

•04518

- -00017

5-1086

•635186

•04573

+ -00038

2-2054

•620890

•04544

+ -00009

2-2096

•619353

•04525

- -oooio

<

Small How.

Mean value, • 05047

8-2446

•381577

•05044

- -00003

8-2446

•381454

•05021

- -00026

5-0887

•388460

•05058

+ -00011

5-0894

•388232

•05066

+ -00019

2-2417

•376414

•04951

- -00096

2-2433

•375879

•04992

- -00055

The observations were taken for different inflow temperatures so as to give approxi¬
mately the same mean temperature for the different values of dO, in consequence of
which I have used the value 4 '2 joules in obtaining the heat watts, the value of S
being the same for the different values of dd. For the large flows, the agreement of
the values of the heat-loss is good, and much better than might be expected, having
obtained the values from experiments made on different dates, although they were
within a day or two of each other, and the calorimeter did not vary more than a few
degrees between each experiment.

The values for the small flow do not agree so well for the 2° rise, but the values for

232

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

the 5° and 8° show that the low7er value given by the former must be exceptional.
Here is an error of nearly 1 part in 1000, to be accounted for by assuming either that
for the small flows the heat-loss is not proportional to the rise in temperature, in
which case the value of the heat-loss per degree rise increases with rise of temperature,
and the value for the 8° rise or the value for the 5° rise must be regarded as
exceptional, or that the error in question was due to some uncertainty at that time in
the experimental conditions. The latter must be regarded as the most probable on
account of the greater difficulty of measuring so small a rise of temperature to the
same order of accuracy. Moreover, the second 15-minute interval show’s a decided
increase, and vrould possibly have attained the correct value given by the mean of
the other readings if the experiment had been further continued. An error of only
'001° on the 2° rise would account for the error in the second interval.

Besides the observations I have just given, which were selected from a series of
trial experiments on the flat heating-wire, a large number of the other experiments
wrere taken with rises of temperature ranging from 1° to 12°. These are detailed in
the tables to be given later, and include results with the central heating- wire as w7ell.
It was a matter of convenience onl}7 that governed my choice of a rise of temperature
for any experiment, and it sometimes happened that it wras more convenient to change
the mean temperature of an experiment by changing the rise of temperature in the
v7ater rather than by altering the inflow temperature — for example, in obtaining a
measure of the specific heat in the neighbourhood of the zero point, where it was
impossible to maintain the inflowing water at a temperature lower than 0° C.

Heat Capacity of the Calorimeter .

Although nearly always negligible in the calculation of results, the thermal capacity
of the calorimeter is of value in showing the size of error introduced by a change in
temperature in the calorimeter water. To determine this, the electrical supply was
suddenly cut off from the calorimeter at a given moment and the rate of fall in tem¬
perature recorded. This was done for both the limits of flow7 used in the present
work. The lag, on breaking the circuit of the thermometer before it commenced to
fall, was in both cases not more than 2 or 3 seconds. If 9 be the temperature indicated
by the outflow7 thermometer above that of the inflow thermometer, then at an}7 time
after shutting off the heat supply, the value of 9 w7ill be approximately 9 =
from which

cl9/dt = hae~at = a9.

But C d9/dt = c/H/di, where C is the thermal capacity of the calorimeter, and H is
the total quantity of heat carried off by the water.

Writing JQd for dW/dt. C a9 = JQ 9, and C = JQ/a.

The followdng set of observations w7as obtained for Calorimeter C : —

BETWEEN THE FREEZING AND BOILING-POINTS.

233

Flow, '598 gramme per second.

Time in seconds, after
breaking electric current.

Temperature of outflow
thermometer.

0

8-375

35

3-968

54

1-721

73

0-574

101

0-237

159

0-008

187-5

0-002

202-5

o-ooo

a — -0386

C = 50 -5 joules.

Flow, '392 gramme per second.

Time in seconds, after
breaking electric current.

l

Temperature of outflow
thermometer.

0

8-068

48

3-646

77

1-400

114

0-264

152

0-142

179

0-032

228

o-oio

266

0-005

299

0-003

320

o-ooo

a= -0311

C = 54 joules.

The logarithmic relation can of course hardly be said to hold with accuracy, or to
be even approximately true for the relation between the fall in temperature and time,
as given by the above series of readings, on account of the sudden descent of the
temperature during the first two minutes. For changes in temperature in the out¬
flowing water, occasioned by a change in the electrical supply or flow, greater than
‘02° during the 15-minute intervals, the thermal capacity as calculated by tbe above
relation is of sufficient accuracy for the application of a small correction. It was
seldom that the variation in temperature of the outflowing water amounted to more
than *02° during a set of readings, and was nearly always less than -01° for the
small flows.

2 H

VOL. CXCIX. — A.

234

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

Sec. 6. — Effect of Stream-line Motion on the Distribution of Heat in the

Fine-flow Tube.

* In Section 2 we discussed two conditions possibly existing in the fine-flow tube of
the calorimeter, and explained generally the effect of these conditions on the
temperature gradient of the glass surface and its influence on the heat-loss. In the
present section I shall give briefly the experimental sequel to the theoretical
considerations. In the light of the recent experiments of Professor Hele-Shaw,*
so beautifully illustrating stream-line motion for water flowing at velocities under the
critical velocity, and of Mr. T. E. Stanton! on 1 The Passage of Heat between
Metal Surfaces and Liquids in Contact with them,’ the results might have been
anticipated which I am about to describe. I do not think, however, that the effect
of stream-line motion in fine tubes has been at all sufficiently appreciated.

As the critical velocity at which the stream-line motion breaks down is so great
(of the order of 10 feet per second) for tubes of from 2 to 3 millims. in diameter, the
effect is inseparably connected with all experiments having to do with tubes of this
size More especially does the flow tend to become linear, and to divide up into
distinct and parallel lines, when a change of viscosity is introduced with a change of
temperature.

I must, in treating this part of the subject, apologise for the present incompleteness
of my experiments, but I feel that I must give such as I have at present, not only to
justify myself for the time and trouble I have taken to completely eliminate the
effect of stream-line motion from my calorimetric measurements, but also as a
beginning to some experiments on the distribution of heat from a metal conductor in
water flowing at different velocities through fine tubes, which I hope to continue in
the near future, and which I hope may at the same time throw some light on the
difference in the rate of flow from the centre to the sides of the tube.

My earlier experiments in 1898 were made with Calorimeter B, with a 2-millim.
bore tube and central heating conductor, but with no special device for preventing
stream-line motion.]; The measurement which we obtained of the mechanical
equivalent at that time, as I have already pointed out, is affected to a certain extent
by this, which was at once apparent when I came to use Calorimeter D, with a
3-millim. bore tube. I undertook two sets of observations with Calorimeter D under
two conditions, one with the heating wire, which was made of six strands of 6-millim.
platinum wire, resting all along the edge of the tube, the other by drawing the wire,
as best 1 could, straight through the centre of the tube.

The results are as follows : —

* ‘Proc. Inst. Naval Arcli.’ (1897), (1898); ‘Proc. Royal Inst/ (1899); ‘ Proc. Liverpool Eng. Soc.’,
20, 37, (1898).

f ‘ Phil. Trans.,’ A, vol. 190, 67 (1897).

1 See note p. 219, supra.

BETWEEN THE FREEZING AND BOILING-POINTS.

235

First Set. — Heating wire resting on sides of fine-bore tube.
Mean Temperature, 26°'l C. November 4, 1898.

do.

Q-

4 • 2 Q d0.

EC.

Difference.

Di ff./dd.

Larc

;e flow.

8-2666

•600605

20-8528

21-6696

•8158

•09870

8-2759

•599645

20-8429

21-6624

•8195

•09902

Small flow.

8-3056

•276991

9-6624

10-3699

•7075

•08518

8-3043

•276888

9-6573

10-3668

•7095

•08544

Second Set. — Heating wire drawn straight through fine-bore tube.
Mean Temperature, 27° C. November 21, 1898.

dd.

Q-

4-2 Q dO.

EC.

Difference.

Di fi./dd.

•

Large

flow.

8-4676

•600527

21-3571

21-6677

•3106

.03668

8-4861

f

•599570

21-3696

21-6695

•2999

•03534

Small flow.

8-8111

•271465

10-0460

10-6749

•6289

•07138

8-8144

•271088

10-0358

10-6724

•6366

•07223

The mean temperature in the two sets is so nearly the same that in comparing the
two we can for the moment neglect the temperature coefficient of the radiation loss
from the glass surface. Without otherwise disturbing the experimental conditions,
the heat-loss for a flow of '60 gramme per second has been reduced exactly one-third
by simply drawing the heating-wire central. The temperature of the outflowing
water being the same in the two cases, the difference in the heat-loss between the
two sets gives a measure of the space represented by the diagram in fig. 2 (p. 154),
between the lines drawn for condition 1 and condition 2.

The value of the heat-loss for the same calorimeter and same flow, but introducing
the device for eliminating stream-line motion, is very nearly '06 watt per degree rise,
which lies midway between '09 and '03 watt as given here respectively. For the
small flow, as might be expected, the heat-loss is more nearly the same in the two

2 h 2

236

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

cases, on account of the greater opportunity for conduction throughout the water
column ; but the effect is still shown. It is of interest also to calculate the actual
temperature of the same heating wire in the two cases.

Flow

per second.

Resistance of
heating wire.

Corrected to a mean
temperature of 26°T.

First set.

•60

•52427 ohm.

•5243

•28

•51740 „

■5174

Second set.

•60

•53460 ohm.

•5331

•27

•52383 „

•5225

Correcting to the same mean temperature by the temperature coefficient of the
platinum, the results show that, as might be expected, the wire held central is
hotter than when in contact with the glass. This means that the central wire
is surrounded by a cloak of hot water moving parallel with it, and the more
completely prevented from diffusing the greater the* velocity of the flow. This is
shown conclusively by comparing the temperature of the wire, as indicated by its
resistance, with the temperature of the same wire measured “ cold ” and reduced to
the same mean temperature. A measurement of this for a current through the wire,
not sufficient to cause a rise in temperature of more than *1° in the outflowing water,
gave the value '5100 ohm.

For the case where the wire is held central in the largest flow and the conditions
are most perfect for the formation of a moving cloak, the mean temperature of the
wire, as given by its increment of resistance, over and above the mean temperature
of the water column in the flow-tube, is of the order of 12° C. This shows that, at
the very most, only one-quarter of the total quantity of water flowing through the
3-millim. flow-tube per second was receiving heat from the wire.

In the case where the wire touches the sides of the tube for its full length, a
greater area of water is heated by conduction and diffusion throughout the layers
along the sides of the tube, which do not move at such a high rate of velocity as in
the centre. The increment of resistance for the large flow given in the first set of
readings for this case shows that the wire was of the order of 7° hotter than the
mean temperature of the water, and indicates that about one-half of the total flow
was employed in carrying off the heat from the wire. When the water is thoroughly
stirred around the heating wire, and in particular where the flat heating wire,
twisted into spirals down the flow-tube, is employed, the mean temperature of the

BETWEEN THE FREEZING AND BOILING-POINTS.

237

wire is much more nearly the same for the two flows and more nearly equal to that
of the water column, the differences being of the order of 1° C. only.

Sec. 7. — Preliminary Measurements of the Mechanical Equivalent.

Our first measurements of the mechanical equivalent in the summer of 1898 were
made with Thermometer C and Calorimeter B. This had a flow-tube slightly less
than 2 millims., and, with the exception of the device for eliminating stream-line
motion, was fitted up in a similar way to the later calorimeters. It is a matter of
interest to determine the way in which the heat-loss varies with rise of temperature
for this case. I have summarized the observations which we made at that time to
determine this, and expressed them here in terms of the same values for the units as
the later measurements. The results are corrected to the same value of Q, and were
all obtained approximately at a mean temperature of 30° C.

Relation of Heat-loss to Rise of Temperature.

Large flow.

Small flow.

Q = ‘54000 gramme per second.

Q = -27300 gramme per second.

cl6.

(EC - 4 • 2 Q, d9)/d6.

dO.

(EC- 4-2 Q d6)/d0.

3-0462

•04445

2-9717

•04941

5 • 9427

•04403

5-8891

•04904

8-9131

•04298

9-0285

•04982

12-2129

•04070

11-9785

•04809

The readings for the large flow are very consistent, as shown by the jflot in fig. 14.
For the small flow the variations in the observations are far from satisfactory, but
they show a similar decrease in the value of the heat-loss with rise of temperature
as for the large flow. The decided bend in the curves shows that, as the temperature
of the out-flowing water is decreased, the temperature gradient down the fine-bore
tube approaches more nearly a straight line (cf. fig. 2, p. 154.) The decrease in the
heat-loss with increase of temperature points to the more perfect confinement of the
heated water around the wire in its passage through the tube, which is occasioned by
its greater difference in density.

The small flow allows of the more perfect distribution of heat throughout the
water column in the flow-tube, and the curve approaches a limiting value, as the
temperature is lowered, much sooner than in the case of the large flow. If we may
assume the two limiting values of the heat -loss per degree rise in the calorimeter for
the two flows by extrapolating for a value of dd — 0 in the two cases, and accept

238

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

these values as being true for any rise where the temperature distribution is uniform
throughout the water column and stream-line motion avoided, then the value of d
may be calculated. The two values of the heat-loss per degree rise so obtained were
for a flow of

'54000 gramme per sec., '04445 watt. '27300 gramme per sec., '04965 watt.
The latter value has not to he corrected for the small conduction effect for the small

flow on account of the method of treatment. Hence the value of d comes out
— '00464, from which

J = 4'2 (1 - 00464) = 4*1805 joules, at 30° C.

The value of J at the same temperature, obtained with the other calorimeters for anv
rise of temperature when using the various devices for obviating stream-line motion,
is 4*1 780 joules, which agrees to 1 part in 2000 with this value. This is quite as
good an agreement as can be expected from the manner of treating the observations,
and the want of agreement in the observations themselves for the small flow, which
is no doubt occasioned by the uncertainty introduced by the stream-lines.

Sec. 8. — Experiments between 0° and 100° C.

As soon as it became clear that the main cause of error had been removed in
eliminating the effect of stream-line motion in the calorimeter, I commenced a series
of experiments to extend over the entire range of temperature. These experiments
are summarized in the following table from I. to LV., and include upwards of 46

BETWEEN THE FREEZING AND BOILING-POINTS.

239

complete experiments. They extended over a period of just a year, and divide
themselves naturally into 8 separate series.'*

Series 1. Nos. I. to XV.

This series includes experiments with Calorimeter C, between 4° and 35° C., and
Calorimeter D, at 28°. Both calorimeters were fitted with a stranded platinum heating-
wire with the silk-covered rubber cord wound round. The distilled water supplied to the
calorimeter was boiled before running under the oil, in bottles forming the head, but
no special care was taken to keep it hot while running in. A large quantity of the
air was driven off1 in the process of boiling, but subsequent results have shown that the
water in the head must still have contained a considerable quantity of dissolved air.
Several of the experiments include other flows, besides the flows used throughout the
entire series of experiments. These have been already summarized in Section 5.
The correction for Thermometer E is that given under Group I., Section 3, c.

Series 2. Nos. XVI. to XXVII.

Between Series 1 and 2 several alterations were made to the apparatus, one of the
chief being the introduction of 40 feet of tin tubing into the constant temperature
tank to replace a similar amount of copper tubing used previously. This was found
necessary owing to the gradual formation of copper rust in the tube. This rust was
carried into the small rubber tube conveying water from the tank to the calorimeter,
and gradually reduced the flow. The experiments were extended from 22° up as far
as 60° C., where they had to be discontinued in order to further refine the regulating-
attachments. The calorimeter was fitted with a solid platinum heating-wire, with
silk-covered rubber cord. The agreement of the results at the lower points between
22° and 35° with those in Series 1 is very satisfactory. Above 45° the results are
not so consistent, probably on account of the fluctuations in gas-pressure supplying
the main heat to the circulating system. The experiment at 60° was taken, however,
when no other gas was being used in the building, and the conditions were unusually
steady. Rises of temperature of 11° and 5° were tried as a check on the measure¬
ments. The correction for Thermometer E is that obtained in the test in Group II.,
Section 3, c.

Series 3. Nos. XXVIII. to XXXII.

This series includes another attempt to obtain the high temperatures, but nearly
all the experiments were spoiled by the liberation of air inside the calorimeter. The
experiment at 67° is given as an illustration of the effect produced by the appearance
of air. In this series the air bubbles were found in the calorimeter water after the
experiment, in spite of the fact that the distilled water was kej3t continuously boiling
as it was supplied to the head bottles, and was cooled from 100° C. only by the cold

* In the following cl is used instead of the 8 of Sec. 2.

240

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

water already in the bottles. The correction to the fundamental interval of Thermo¬
meter E is the same as for the last series.

Series 4. Nos. XXXIII. to XXXV.

This series includes experiments with Calorimeter C, using the flattened platinum
strip for heating wire, instead of the central wire conductor and elastic strip. These
observations are summarized in Section 5. The use of the flat wire was found to
produce more irregularity in the heat-loss between the different flows, especially
apparent on the small flows, and no doubt occasioned by the fact that the wire
touched the sides of the tube. The correction for the fundamental interval of
Thermometer E is that given under Group III., in Section 3, c.

The agreement of the value of J with the other measurements is very satisfactory,
and the more so because the heating conductor was changed not only in form but in
resistance.

Series 5. Nos. XXXVI. and XXXVII.

Ex]i>eriments with Calorimeter E with flat heating strip. The fine-bore tube was
slightly over 3 millims. in diameter, and was the largest tried in these experiments.
The first experiments with this calorimeter were made with the central wire conductor
and large rubber cord, but were neither satisfactory nor consistent. The effect of the
streamdine motion apparently began to come in, with the helical motion in the water,
probably from the size of the flow-tube compared to the size of the heating-wire.
The thermometer was the same in every respect to that used in Series 4.

Series 6. Nos. XXXVIII. to XLVIII.

In this series, Calorimeter C was refitted with central solid platinum heating-wire
with silk-covered rubber cord. Thermometer E, Group III., was still used. Measure¬
ments from 20° to 0° were made, and the values obtained under Series 1 and 2
completely verified. The observations were extended below 4°, and the lowest point
obtained was for a rise of temperature of 1° above 0°. In this experiment only one
flow was obtained, but the value of d may be calculated with some degree of
approximation by assuming the value of the heat-loss for the two other determina¬
tions with inflow-water at 0°, and correcting for the temperature coefficient of the
radiation. This experiment was done principally to test the rapid increase of the
specific heat at the freezing-point. An attempt was also made to obtain the high
points, but with no more success than in the previous attempts.

Series 7. Nos. XLIX and L.

Between Series 6 and the present series, various devices were tried to obviate the
effect of dissolved air in the boiled water. The plan was finally adopted of preparing

BETWEEN THE FREEZING AND BOILING-POINTS.

241

absolutely air-free water, as described in Section 5, and from this time on the work at
the higher points progressed more favourably. In this series observations were
obtained at 30° and 86°, but the latter unfortunately with only one flow. The second
flow could not be taken on account of the rapid evaporation of water from the tank,
the constant-level device not being used at that time. Here again, extrapolation for
the heat-loss from the value at 30° can be made, but the procedure can hardly be
justified with a greater accuracy than 1 in 1000. The agreement with the later
results is, however, extraordinarily good.

Series 8. Nos. LI. to LV.

This series was made with box 2 and Thermometer E, involving the F.I. correction
in Group IV. It extends from 32° C. to 92° C., and in many respects is the most
important series of the whole. The Calorimeter C was refitted throughout so as to
give an entirely new set of observations. The complete agreement of the measure¬
ment of the specific heat at 32° C. with the other measurements with box 1, eliminates
any possible error due to the box and its connections. The four sets of observations
at the higher points are exceedingly consistent, and distinctly show that the previous
trouble to obtain the measurements in this region was due to the effect of the air in
the water. Not the slightest trouble was experienced with air making its appearance
in the calorimeter in these experiments. The order in which these observations were
taken was as follows : 32°, 74°, 92°, 80° and 68°, between which the calorimeter
cooled down to the temperature of the laboratory, and had to be heated up to the
desired point each time. The measurement at 86° in Series 7 is in very good agree¬
ment with these. The beautiful consistency of this last series of measurements might
make it desirable to repeat the observations between 50° and 60° C. with air-free
water. I did not consider this was necessary, however, as the continuity of the
observations at the two ends of the range is so good, and the divergence in the results
obtained between 50° and 60° C. is so clearly explained by unforeseen and extraneous
causes.

The calculation of the results in the tables just given for the determination of the
value of the electrical and heat energy has been very much facilitated by the use of
the Brunsviga calculating machine, which is very much to be recommended for this
class of work. For the application of the small correction factors, and for the final
estimation of the values of d and h, the Fuller cylinderical slide-rule has been
constantly used. The values of the electrical and heat watts given in the summary
at the foot of each table may therefore be in error by 2 or perhaps 3 in the fourth
decimal place, but no more, but I feel confident that in the estimation of the mean
value, upon which the value of d depends, this error tends to disappear, and that the
value of d given by the measurements in any of the tables, represents the observa¬
tions to an accuracy of 1 part in 100,000.

VOL. CXCIX.— A. 2 I

242

DR. H. T. BARNES ON TIIE CAPACITY FOR HEAT OF WATER

Summary of Results of Observations.

The following table contains a summary of the observations arranged and
abstracted by Professor Callendak. The original tables (abridged), giving details
of corrections and calculations, are preserved for reference in the Archives of the
Royal Society.

The first column gives the number and date of the corresponding Abridged Table
preserved in the Archives. The second column gives the temperature of the jacket
water or inflow, taken by means of a mercury thermometer, and corrected to the
nearest hundredth of a degree. The third column gives the mean difference of
temperature, (W, between the inflow and outflow, observed to the ten-thousandth of a
degree by means of a pair of differential platinum thermometers, and reduced by the
parabolic difference formula, assuming the boiling-point of sulphur to be 444'53° C.
The fourth column gives the flow of water, Q, through the calorimeter, in grammes
per second, reduced to vacuum. The fifth column gives the value of the product,
4'2 Q dd, for comparison with the power, EC in watts, given m the next column.
The seventh column gives the difference, EC — 4'2 Q dd, of the numbers in the two
previous columns divided by dd. This quotient is denoted by I), and is used in
calculating the results given in the last column, by means of the difference equation

EC /dO — 4 2 Q = D = 4-2 Q d + h,

in which d expresses the fractional variation of the specific heat of water in terms of
an arbitrary unit 4 '200 joules, as defined by the relation J = 4 '200 (1 -f- d), and
the symbol h denotes the rate of heat-loss in watts per degree rise of temperature.
The value of d is found by combining the observations for the two different flows Q'
and Q", which give the relation

d = (D' - D")/4-2 (Q' - Q").

The values of h and J follow immediately from that of d. The values of EC and J
are calculated assuming the E.M.F. of the Clark cell at 15c C. to be 1'4342 volts,
but this does not affect the relative values.

In cases where more than two different flows were taken at the same temperature,
the values of d and h are calculated from the largest and smallest flow. These
values of d and h are then assumed to calculate a value of D for the intermediate
flow for comparison with the value of D deduced from the observations.

BETWEEN THE FREEZING AND BOILING-POINTS.

243

Table XVIII. — Summary of Results of Observations.

Number
and date.

Jacket.
Temp. ° C.

Temp. rise.
dd, ° C.

Flow Q.

gm./sec.

Product.
4-2 Q dd.

Watts.

EC.

DifF ./dd.

D.

Results.
d, h, J.

Series I. Calorimeter D. Stranc

led Conclucto

r. Therm om<

jter E, I.

I.

24° 03

7 °5234

•674106

21-3006

21-7331

•05749

d= - -00485,

Feb. 15,

(Not re

peated)

h = + -07122,

1899.

24-04

7-7745 ..

•399290

13-0379

13-5290

•06317

J= 4-1796,

24-02

7-9463

•390196

13-0225

13-5245

•06318

at 28° -01 C.

II.

24-00

7-8882

•496655

16-4544

16-9383

•06135

d= - -00533,

Feb. 15,

(Not re

peated)

h= + -07247,

1899.

24-01

8-0033

•248234

8-3447

8-8802

•06691

J= 4-1776,

(Not re

peated)

at 28° -01 C.

Calorimeter C.

Stranded Conductor. Thermometer E, I.

III.

25-03

7 -9646

■666042

22-2800

22-5697

•03637

d= - -00469,

Feb. 20,

25-03

7-9775

•664388

22-2607

22-5504

•03632

h= + -04944,

1899.

25-06

8-2608

•398498

13-8260

14-1689

•04151

J= 4-1803,

25-06

8-2560

• 398540

13-8194

14-1634

■04167

at 29° -10 C.

IV.

25-03

8-2281

•258114

8-9199

9-2922

•04525

d= - -00546,

Feb. 20,

25 • 03

8-2284

•257947

8-9144

9-2879

■04539

h= + -05123,

1899.

25 • 06

8-2199

•501957

17-3294

17-6567

•03982

J= 4-1771,

25-06

8-2301

•501026

17-3186

17-6450

•03966

at 29° -09 C.

V.

25-07

7-9031

•660813

21-9361

22-2194

•03585

d = - -00489,

Feb. 22,

25-07

7 • 9083

•658690

21-8800

22-1628

•03576

h= + -04937,

1899.

,25-07

8-2680

•392575

13-6334

13-9749

•04131

J= 4-1795,

25-07

8-2635

•392362

13-6280

13-9692

•04129

at 29° -11 C.

VI.

25 • 07

8-1938

•496670

17-0937

17-4137

•03906

D calc.

Feb. 22.

25-07

8-1844

•496553

17-0700

17-3911

•03924

= -03917.

VII.

25-01

8 • 3390

•590477

20-6807

20-9912

•03723

d= - -00499, I

Mar. 2,

25-01

8-3439

•589356

20-6536

20-9653

•03734

h= + -04965,

1899.

25-01

8-4310

•375154

13-2841

13-6363

•04177

J= 4-1790,

25-01

8-4304

•375076

13-2807

13-6362

•04181

at 29° -21 C.

VIII.

25-01

8-3979

•472489

16-6652

16-9992

•03977

D calc.

Mar. 2.

25-01

8-4060

•471670

16.6524

16 • 9885

•03998

= -03975.

IX.

9-51

8-5768

•372746

13-4272

13-7377

•03620

d= - -00208,

Mar. 9,

9-51

8-5803

•372262

13-4153

13-7261

•03622

h = + -03946,

1899.

9-51

8-5499

•573318

20-5876

20-8827

•03452

J= 4-1913,

9-51

8"5616

•571920

20-5654

20-8595

•03439

at 13° -79 C.

X.

9-51

8-5586

•459149

16-5046

16-8060

•03522

D. calc.

Mar. 9.

9-51

8-5683

•458194

16-4889

16-7912

•03529

= -03546.

XII.

13-30

8-7411

•357638

13-1298

13-4508

•03673

d= - -00309,

Mar. 11,

13-30

8-7241

•358131

13-1222

13-4428

•03675

h = + -04139,

1899.

13-31

8-9195

•566221

21-2117

21-5153

•03404

J= 4-1870,

(Second in

terval, regu

lator failed)

at 1 7 J • 6 9 C.

2 I 2

244

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

Table XVIII. — Summary of Results of Observations — continued .

Number
and date.

Jacket.
Temp. °C.

Temp. rise.

dd, 0 c.

Flow Q.
gm./see.

Product.
4-2 Q dd.

Watts.

EC.

T>ie./dd.

D.

Results.
cl, h, J.

Series I.

Calorimeter

C. Stranded

Conductor.

Thermometer E, I. — cont

inued.

XIII.

19-95

8°-4978

•373778

13-3404

13-6769

•03960

d= - -00446,

Mar. 16,

19-95

8-5149

•372934

13-3371

13-6734

•03950

h= + -04655,

1899.

19-95

8-4695

•587077

20-8834

21-1846

•03556

J= 4-1813.

19-95

8-4963

•584879

20-8686

21-1708

•03557

at 24° -20 C.

XIV.

30-15

8-6230

•358452

12-9819

13-3758

•04568

a 1= - -00561,

Mar. 17,

30-16

8-6154

•358388

12-9682

13-3639

•04593

h= + -05424,

1899.

30-16

8-6037

•567099

20-4924

20-8450

•04099

J= 4- 1765.

30-17

8-6283

•565123

20-4794

20-8314

•04080

at 34° -47 C.

XV.

0-13

8-3170

•559701

19-5511

19-9014

•04212

d= + -00330,

Mar. 24,

0-13

8-3407

•557801

19-5402

19-8913

•04210

h= + -03437,

1899.

0-13

8-2833

•361401

12-5730

12-8992

•0393S

J= 4-2138.

0-13

8-2883

•360873

12-5622

12-8898

•03953

at 4°-28C.

Series II. Calorimeter C. Solid Conductor.

Thermometer E, II.

XVI.

18°- 17

7-9225

•404339

13-4542

13-7653

•03927

d= - -00413,

June 6,

18-16

7-9999

•633704

21-2922

21-5737

•03519

h = + -04619,

1899.

18-17

7-9672

•401420

13-4324

13-7441

•03913

J= 4-1827,

at 22° -16 C.

XVII.

27-29

8-2190

•618698

21-3573

21-6793

•03918

d= - -00542,

June 8,

27-30

8-2310

•617503

21-3472

21-6712

•03936

h= + -05334,

1899.

27-30

8-1357

•394624

13-4843

13-8447

•04430

J= 4-1773,

27-30

8 ■ 1545

•393530

13-4780

13-8404

•04444

at 31° -40 C.

XVIII.

27-97

8-4281

•602990

21-3447

21-6740

•03908

d= — ‘00536,

June 9,

27 • 97

8-4467

•601265

21-3305

21-6632

•03939

h= + -05282,

1899.

27-97

8-3590

• 383786

13-64739

13-8410

•04392

J= 4-1775,

27-97

8-3803

•382529

13-4640

13-8360

•04439

at 32° -17 C.

XIX.

36-82

8-3641

•636775

22-3694

22-7438

•04476

d= - -00540,

■ June 12,

36-82

8-3726

•635489

22-3469

22-7246

•04511

/; = + -05939,

1899.

36-83

8-3746

•381973

13-4352

13-8606

•05068

J= 4-1773,

36-84

8-3717

•381800

13-4245

13-8493

•05074

at 41° -02 C.

XX.

41-30

8-3720

•632463

22-2389

22-6538

•04956

d= - -00514,

i June 12,

41 ■ 30

8-3892

•630617

22-2195

22-6331

•04931

h= + ' 06306,

1899.

41-30

8-4063

•397132

14-0213

14-4785

•05439

J= 4-1784,

41-30

8-4068

•396628

14-0043

14-4633

•05460

at 45° -49 C.

XXI.

45-70

7-9687

•666393

22-3033

22-7174

•05197

d= - -00531,

June 14,

45-70

7-9689

•665414

22-2705

22-6825

•05171

h= + -06669,

1899.

45-69

7-9639

•419567

14-0339

14-4908

"05737

J = 4 - 1777,

45-70

7-9648

•418914

14-0136

14-4700

•05730

at 49° -68 C.

XXII.

50'52

8-2051

•631248

21-7537

22-2358

•05876

d= - -00372,

June 14,

50-52

8-2065

•630085

21-7173

22-2001

•05884

h= + -06867,

1899.

50-50

8-1496

•401674

13-7486

14-2543

•06206

J= 4-1844,

50-53

8-1528

•400859

13-7261

14-2374

•06272

at 54° -61 C.

BETWEEN THE FREEZING AND BOILING-POINTS.

245

Table XVIII. — Summary of Results of Observations — continued.

Number
and date.

Jacket.
Temp. ° C.

Temp. rise.
dO, ° C.

Flow Q.
gm./sec.

Product.
4-2 Q dd.

Watts.

EC.

Diff./dd.

D.

Results.
d, h, J.

Series II.

Calorimete

;r C. Solid C

conductor. T

hermometer !

1

3, II. — continued.

XXIII.

55° 64

8 °3805

•612400

21-5553

22-0876

•06351

d= - -00341,

June 17,

55 • 64

8 • 3835

•611227

21-5217

22-0532

■06339

h= + -07220,

1899.

55-61

8-3534

•388491

13-6299

14-1874

■06674

J = 4-1849,

55-61

8-3674

•387534

13-6191

14-1760

•06656

at 59° -80 C.

XXIV.

55-61

8-3158

•462971

16-1699

16-7134

•06536

D calc.

June 17.

55-61

8 • 3395

•461364

16-1595

16-7052

■06542

= -06559.

XXV.

27-15

11-3447

•642348

30-6064

31-0604

•04002

d= - -00528,

June 20,

27-15

11-3383

•641338

30-5410

30-9955

•04008

h = + -05429,

1899.

27-17

11-3324

•402771

19-1703

19-6835

•04529

J= 4-1778,

27-17

11-3428

•402108

19-1562

19-6715

•04542

at 32° -81 C.

XXVI.

27-16

11-2504

•489797

23-1437

23-6345

•04363

D. calc.

June 20.

27-17

11-2418

•488845

23-0811

23-5726

•04372

= -04345.

XXVII.

27-98

5-1297

•610593

13-1551

13-3565

•03926

d= - -00555,

June 22.

27-98

5-1365

•608831

13-1344

13-3379

•03962

h= + -05364,

27-99

5-1579

•383538

8-3086

8-5392

•04471

J= 4-1767,

27-99

5-1613

•382650

8-2948

8-5256

■04472

at 30° -54 C.

Series III. Calorimeter C. Thermometer E,

II. Solid Conductor.

XXVIII.

25° 44

8° 0202

•634819

21-3838

21-7067

•04026

d= - -00496,

Sept. 4,

, 25-44

8-0343

•632988

21-3596

21-6841

•04039

h = + -05352,

1899.

25-45

8-0981

• 384855

13-0896

13-4580

•04549

J= 4-1792,

25-45

8-1079

•384098

13-0797

13-4487

•04552

at 29° -47 C.

XXIX.

27-14

8-0488

•630942

21-3290

21-6566

•04070

d= - -00544,

Sept. 6,

27-14

8-0548

•630005

21-3131

21-6420

•04083

h = + -05518,

1899.

27-14

8-1703

•380614

13-0608

13-4412

•04656

J= 4-1771,

27-14

8-1808

•379886

13-0526

13-4340

•04662

at 31° • 22 C.

XXX.

34-60

8-3187

•394336

13-7775

14-2055

•05145

d= - -00530,

Sept. 12,

34-60

8-3139

•394088

13-7609

14-1916

■05180

h= + -06041,

1899.

34-60

8-3018

•626817

21-8555

22-2412

•04647

J= 4-1777,

34-60

8-3159

■625193

21-8360

22 • 2225

•04648

at 38° -76 C.

XXXI.

63-33

8 • 3443

•390990

13-7027

14-3302

•07520

d= - -00395,

Sept. 14,

63 • 33

8-3517

•390431

13-6952

14-3256

•07547

h= + -08182,

1899.

63 • 33

8-3782

•627514

22-0812

22-6775

•07117

J= 4-1834,

63-33

8-4042

•625068

22-0633

22-6657

•07168

at 67° -52 C.

XXXII.

50-41

8-2923

•393061

13-6893

14-2326

•06552

d= - -00473,

Sept. 18,

50-42

8-2990

•392595

13-6841

14-2267

•06538

h= + -07325,

1899.

50 • 43

8-2824

•626272

21-7855

22-2875

•06061

J= 4-1801,

50-43

8-2965

•624643

21-7658

22-2722

•06104

at 54° -57 C.

246

1)R. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

Table XVIII. — Summary of Results of Observations — continued.

Number
and date.

Jacket.
Temp. ° C.

Temp. rise.

de;° c.

Flow Q.

gm./sec.

Product.
4-2 Q dQ.

Watts.

EC.

Di ff./dd.

D.

Results.
d, h, J.

Series IV.

Calorimet

sr C. Spiral

strip, no rubl

>er cord. Th

milometer E

1, III.

XXXIII.

24° 64

8 -°3069

•626436

21-8557

22-2313

•04521

d= - -00493,

Oct. 1 4,

24-64

8-3212

•625128

21-8476

22 • 2248

•04533

h= + -05826,

1899.

24 • 65

8-2446

•381577

13-2130

13-6288

•05044

J= 4-1793,

24 • 65

8-2446

•381454

13-2087

13-6226

•05021

at 28° -77 C.

XXXIV.

25-97

5-1009

• 636545

13-6372

13-8676

•04518

d= - -00497,

Oct. 18,

25-97

5-1086

•635186

13-6286

13-8622

•04573

h= + -05870,

1899.

25-98

5-0887

•388460

8-3024

8-5598

•05058

J= 4-1791,

25-98

5-0894

•388232

8-2987

8-5565

•05065

at 28° -52 C.

XXXV.

27-38

2-2053

•620890

5-7508

5-8510

•04544

d= - -00427,

Oct. 19,

27-38

2-2095

•619353

5-7475

5-8475

•04525

h— + -05645,

1899.

27-38

2-2416

•376414

3-5437

3-6547

•04951

J= 4-1821,

27-38

2-2432

•375879

3-5412

3-6532

•04992

at 28° -49 C.

Series

V. Calorimeter E (3 millims.), spiral strip. Thermometer E, III.

XXXVI.

25° 74

8° -3281

•623288

21-8013

22-1831

•04584

d= - -00542,

Oct. 27,

25-75

8-3360

•622427

21-7919

22-1727

•04566

h= + -05994,

1899.

25-74

8-3310

•384073

13-4387

13-8652

•05119

J= 4-1772,

25-75

8-3436

■383177

13-4377

13-8551

■05122

at 29° -92 C.

XXXVII.

17-02

7-7440

•689271

22-4184

22-7443

•04208

d= - -00404,

Nov. 1.

17-02

7-7488

•687919

22-3883

22-7199

•04280

h= + -05414,

17-08

7-7804

•485563

15-8671

16-2241

•04589

J= 4-1830,

(Flow no

t repeated)

at 20° -92 C.

Series VI. Solid Conductor.

Thermometer E, III. Calorimeter C.

| XXXVIII.

16° 01

8-2408

•603561

20-8901

21-1857

•03587

d= - -00384,

Nov. 3,

16-01

8-2549

•602250

20-8804

21-1744

•03598

h = + -04567,

1899.

16-01

8 • 2575

•375191

13-0122

13-3381

•03947

J= 4-1838,

16-01

8-2554

•375065

13-0043

13-3325

•03976

at 20° -18 C.

i xxxix.

16-00

2-1866

•576497

5-2944

5-3737

•03627

d = - -00310,

Nov. 4,

16-00

2-1878

•575717

5-2901

5-3699

•03647

h= + -04387,

1899.

16-00

2-1855

•357957

3-2857

3-3713

•03917

J= 4-1870,

16-00

2-1866

•357600

3-2841

3-3699

•03924

at 17° -09 C.

XL.

17-57

5-0878

•606172

12-9531

13-1370

•03615

d = - -00397,

Nov. 6,

17-57

5-0992

•604361

12-9433

13-1274

•03611

h= + -04622,

1899.

17-59

5-1071

•377173

8-0903

8-2943

•03995

J= 4-1833,

17-60

5-1142

•376515

8-0874

8-2916

•03993

at 20° -13 C.

XLI.

16-34

8-1889

•608476

20-9275

21-2381

•03793

d= - -00420,

Nov. 14,

16-34

8-2003

•607098

20-9093

21-2208

•03799

h= + -04867,

1899.

16-36

8-2198

•383107

13-2261

13-5703

•04188

J= 4-1824,

16-38

8-2261

•382518

13-2159

13-5613

•04199

at 20° -45 C.

BETWEEN THE FREEZING AND BOILING-POINTS.

247

Table XVIII. — Summary of Results of Observations — continued.

Number
and date.

Jacket.
Temp. ° C.

Temp. rise.
dO, ° C.

Flow Q.
gm./sec.

Product.

4-2 Q dd.

Watts.

EC.

Diff ./dd.

D.

Results.
d, h, J.

Series VI.

Solid Conductor. Thermometer E, I

[I. Calorime

ter C. — cont

inued.

XLII.

14-65

2-1581

•581584

5-2715

5 ■ 3533

•03790

d= - -00254.

Nov. 16,

14-65

2-1614

•579646

5-2620

5 • 3464

•03905

h= + -04467,

1899.

14-63

2-1391

•363655

3-2672

3-3543

•04072

J= 4-1893,

14-63

2-1420

•362870

3-2646

3-3521

•04085

at 15° -71 C.

XLIII.

7-50

2-1065

•587245

5-1955

5-2791

•03969

d = + -00022,

Nov. 17,

7-57

2-1050

•586733

5-1873

5-2720

•04024

h = + -03943,

1899.

7-67

2-0664

•372448

3-2325

3-3141

•03949

J= 4-2009,

7-70

2-0577

•372092

3-2158

3-2982

•04044

at 8° -66 C.

XLIV.

0-15

2-1993

•592582

5-4737

5-5842

•05024

d= + '00512,

Nov. 18,

0-15

2-1982

•591463

5-4606

5-5714

•05041

h= + -03759,

1899.

0-15

2-1909

•389629

3-5853

3-6861

•04601

J= 4-2215,
at W35 C.

0-15

2-1916

•389077

3-5813

3-6820

•04594

XLY.

0-15

5-0561

•616583

13-0935

13-3381

•04838

d= + -00370,

Nov. 22,

0-15

5-0694

•614942

13-0929

13-3344

•04764

h= + -03842,

1899.

0-15

5-0738

•369790

7-8802

8-1040

•04410

J= 4-2155,

0-15

5-0772

•369335

7-8758

8-1007

•04429

at 2° -68 C.

XLYI.

0-15

1-0483

•598234

2-6339

2-6880

•05160

d= + -00597,

Nov. 22.

0-15

1-0493

•597057

2-6313

2-6863

■05241

h= + -0370.

XL VII.

25 • 22

8-4686

•601120

21-3807

21-7224

•04035

d= - -00497,

Nov. 27,

25 • 22

8-4910

•599162

21-3675

21-7149

•04093

h = + ‘05316,

1899.

25-21

8-4064

•377559

13-3304

13-7105

•04522

J= 4-1791,

25-21

8-4159

•376851

13-3205

13-7023

•04537

at 29° -43 C.

1 XL VIII.

46-77

8-5495

•592608

21-2822

21-7592

•05579

d= - -00513,

Nov. 29,

46-78

8-5767

•590139

21-2580

21-7367

• 05582

h= + -06855,

1899.

46-79
(Not re

8-4380

peatecl)

•370080

13-1155

13-6267

•06058

J= 4-1785,
at 51° -02 C.

Series VII.

Calorimeter

C. Solid Conductor. Thermometer E,

III. Air-free Water.

XLIX.

24-94

8 -3727

•579770

20-3878

20-7297

•04083

d=* -00488,

Jan. 1,

24-94

8-3774

•579087

20-3751

20-7219

•04140

h = + -05299,

1900.

24-95

8-3435

•355436

12-4554

12-8365

•04567

J= 4-1795,

24-95

8-3449

•355120

12-4465

12-8282

■04574

at 29° -13 C.

L.

81-56

8-0278

•646259

21-7898

22-5230

•09133

d= - -00052,

Jan. 6.

81-60

8-0325

■ 644355

21-7383

22-4762

•09186

h= + -0930.

Series VIII. Calorimeter C. Solid Conductor.

Thermometer E, IV. Box II.

LI.

1 28 • 15

8-2035

•645683

22-2468

I 22-6039

•04358

d= - -00521,

Feb. 24,

28-15

8-2223

•643769

22-2317

22-5931

•04396

h = + -05791,

1900.

28-16

8-2468

•404207

14-0003

14-4041

•04897

J = 4-1781,

1

28-16

8-2613

•403213

13-9904

14-3966

•04917

at 32° • 26 C.

1

248

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

Table XVIII. — Summary of Results of Observations — continued.

Number
and date.

Jacket.
Temp. ° C.

Temp. rise.
d.9, c C.

Flow Q.
gm./sec.

Product.
4-2 Q d6.

Watts.

EC.

DiS./dd.

D.

Results.
d, h, J.

Serie

s VIII. Calorimeter C

Solid Cone

uetor. Then

ammeter E, T

V. Box II.-

—continued.

LII.

69 °85

8-°3152

•622088

21-7257

22-3864

•07947

d= - -00189,

Feb. 28,

69-85

8-3226

• 620830

21-7011

22-3672

•08044

h= + -08469,

1900.

69-86

8-4472

•385739

13-6853

14-3733

•08146

J= 4-1920,

69-86

8-4893

•383723

13-6816

14-3761

•08181

at 74’ -05 C.

LIII.

87-42

8-2361

•645077

22-3142

23-1450

•10088

d= + -00042,

Mar. 10,

87-41

8-2513

•642851

22-2783

23-1161

• 10155

h= + -10011,

1900.

87-43

8-2768

•402934

14-0069

14-8450

•10130

J= 4-2017,

87-43

8-3168

•400935

14-0049

14-8398

•10030

at 91° ‘55 C.

LIV.

76-12

8-5262

•617742

22-1213

22-8806

•08894

d=- -00117,

Mar. 17,

76-12

8-5570

•615198

22-1099

22-8747

•0S938

li= + -09218,

1900.

76-12

8-5224

•388311

13-8992

14-6692

•09036

J= 4-1951,

76-12

8-5433

•387095

13-8897

14-6603

•09021

at 80’ ■ 38 C.

LY.

63-84

8 • 6883

•387767

14-1499

14-8324

■07856

d= - -00262,

Mar 21,

63-84

8-6760

•387527

14-1212

14-8029

•07858

h = + -08283,

1900.

63-82

8-8222

•604663

22-4047

23-0757

•07606

J= 4-1890,

63-82

8-8494

■601956

22-3732

23-0488

•07634

at 68° -21 C,

Each single line in the above table represents the mean results of the observations
of temperature and potential difference taken, as explained and illustrated by the
specimen tables of observations, during a period of 15 minutes, for which the corres¬
ponding value of the flow was measured. In nearly all cases the observations were
repeated during a second period of 15 minutes under conditions as nearly as possible
the same, except for a slight diminution of the flow, due to the fall in the water
level. The order of accuracy of the readings can be estimated by comparing the
corresponding values of D for the two similar flows. The two values of D should
agree, except that the falling-off of the flow tends to make the second value in each
case slightly the larger when d is negative. In comparing the values it must be
remembered that 3 in the fifth place of D corresponds to only 1 part in 100,000, with
a heat supply of 24 watts and a rise of 8°. The differences seldom exceed 1 in f 0,000,
whereas with the method of mixtures it is very difficult to obtain an order of
agreement of 1 in 1,000 in repeating an experiment under identical conditions.

The values of d as directly measured and expressed in the different series just
given, I have plotted in fig. 17. From the smooth curve drawn so as to include
the observations, I have taken the following values of d and calculated the
corresponding values of J. These are summarized here (p. 250). The values of J
are, of course, in absolute measure, and the values of d in terms of a thermal
unit equal to 4-2000 joules, which occurs at 9° C. and 88 '5 C. The mean value of

BETWEEN THE FREEZING AND BOILING-POINTS.

249

J, 4*18876 joules, is exactly coincident with the values at 15°*7 and at 68° C. In
selecting a thermal unit to which the values of the specific heat may be referred,
it seems desirable to adopt one at a temperature which, if at the same time at a
convenient part of the scale, may also represent the mean value over the whole
range. Such a convenient point appears to he indicated at either 15° or 16° C.
T propose, at the present time, to adopt the value at 16° C., and shall in conse¬
quence express the specific heat of water in terms of this unit, which is equal to
4*1883 joules, and which differs from the mean value by only 1 part in 10,000.

The following table (p. 252) includes the values of the specific heat of water in
terms of a unit at 16°. No one simple formula can be fitted to the complete curve
between 0° and 100° with any degree of accuracy, on account of the change which
occurs at 37°*5, which is the point of minimum specific heat. Two formulae can be

Fig. 17.

fitted together, however, over the range with great accuracy. Between 5° and 37°*5
the expression representing the specific heat in terms of a unit at 1 6° C. is

S* = *99733 + *0000035 (37*5 - tf + *00000010 (37*5 - tf.

The same expression reads above the minimum point, as far as 55°, in this form :

S* = *99733 + *0000035 (t - 37*5)2 + *00000010 (t - 37*5)3.

At 55° and upwards the values diverge more and more from this formula, and follow
VOL. CXCIX. — A. 2 K

250

DE. H. T. BARNES ON THE CAPACITY FOE HEAT OF WATEE

a different curve. The slope of the curve above 60° is very nearly the same as the
well-known formula of Regnault, but the rate of increase is very much smaller.
The following expression, which is very nearly a linear relation, holds between
50° and 100° :

S, = -99850 + -000120 ( t - 55°) + ’00000025 ( t - 55)2.

These two formulae I have represented in the last table in column 3. They represent
the variation of the specific heat of water very clearly with the exception of the
rapid increase at 0°, and are entirely independent of the values assigned to my
electrical units. They can be changed to fit a unit at any other temperature by
simply changing the constant term.

Since it would be a matter of great labour to determine the specific heat of
superheated water, and since the variation curve of the specific heat shows no
discontinuity as the boiling-point is reached at atmospheric pressure, this last formula
may be said to hold with some claim to accuracy above 100° throughout the entire
range covered by Regnault’s experiments.

Comparison of Observed and Calculated Values.

Temperature.

Observed cl.

Calculated cl.

J observed.

J. calculated.

Series

|

I. February to March, 1899.

1

28-01

- -00485

- -00509

4-1796

4-1786

29-09

- -00469

- -00517

4-1803

4-1783

29-11

- -00489

- • 00517

4-1795

4-1783

29-21

- -00499

- -00519

4-1790

4-1782

13-79

- -00208

- -00210

4-1913

4-1912

17-69

- -00309

- -00328

4-1870

4-1862

24-20

- -00446

- -00460

4-1813

4-1807

34-46

- -00561

- -00545

4-1765

4-1771

4-28

+ -00330

+ -00310

4-2138

4-2130

Series II. January 6-22

1899.

22-16

- -00413

- -00425

4-1827

4-1822

31-40

- -00542

- -00532

4-1773

4-1777

'32-17

- -00536

- -00536

4-1775

4-1775

41-02

- -00540

- -00539

4-1773

4-1774

45-49

- -00514

- -00515

4-1784

4-1784

49-68

- -00531

- -00480

4-1777

4-1799

54-61

- -00372

- -00430

4-1844

4-1819

59-80

- -00341

- -00370

4-1849

4-1845

32-81

- -00528

- -00540

4-1778

4-1773

30-54

- -00555

- -00528

4-1767

1

4-1779

BETWEEN THE FREEZING AND BOILING-POINTS.

251

Comparison of Observed and Calculated Values — continued.

Temperature.

Observed d.

Calculated d.

J. observed.

J. calculated.

1

Series III. September 4-1

8, 1899.

29-47

- -00496

- -00520

4-1792

4-1782

31-22

- -00544

- -00530

4-1771

4-1778

38-76

- -00530

- -00545

4-1777

4-1771

67-52

- -00395

- -00275

4-1834

4-1885

54-57

- -00473

- -00430

4-1801

4-1820

Series IV. October 14-19

, 1899.

28-77

- -00493

- -00515

4-1793

4-1784

28-52

- -00497

- -00510

4-1792

4-1786

Series V.

October 27 to November 1, 1899.

29-92

- -00542

- -00523

4-1772

4-1781

20-92

- -00404

- -00404

4-1830

4-1830

Series VI. November 3-29, 1899.

20-18

- -00383

- -00384

4-1838

4-1839

17-09

- -00310

- -00310

.4-1870

4-1870

20-13

- -00397

- -00383

4-1833

4-1839

20 • 45 '

- -00420

- -00400

4-1824

4-1832

15-71

- -00254

- -00270

4-1893

4-1887

8-66

+ • 00022

+ -00020

4-2009

4-2008

1-35

+ -00512

+ -00560

4-2215

4-2235

2-68

+ -00370

+ -00430

4-2155

4-2181

29-43

- -00497

- -00520

4-1791

4-1782

51-02

- -00513

- -00470

4-1785

4-1803

Series VII. January 1-6,

1900.

29-13

- -00488

- -00517

4-1795

4-1783

85-60

- -00052

- -00036

4-1978

4-1985

Series VIII. February 24 to March 21, 1900.

32-26

- -00521

- -00535

4-1781

4-1776

74-05

- -00189

- -00189

4-1920

4-1920

91-55

+ -00042

+ -00042

4-2017

4-2017

80-38

- -00117

- -00110

4-1951

4-1954

68-21

- -00262

- -00270

4-1890

4-1887

Time of flow, 900 seconds automatically recorded.

2 k 2

252

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

Summary of the Specific Heat of Water from Smoothed Curve.

Temperature ° C.

d.

J.

o

5

+ -00250

4-2105

10

- -00050

4-1979

15

- -00250

4-1895

20

- -00385

4-1838

25

- -00474

4-1801

30

- -00523

4-1780

35

- -00545

4-1773

40

- -00545

4-1773

45

- -00520

4-1782

50

- -00480

4-1798

55

- -00430

4-1819

60

- -00370

4-1845

65

- -00310

4-1870

70

- -00245

4-1898

75

- -00180

4-1925

80

- -00114

4-1954

85

- -00043

4-1982

90

+ -00025

4-2010

95

+ -00090

4-2038

Mean value ....

4-18876

Variation of the Specific Heat of Water in Terms of a Thermal Unit at

16° C. = 4-1883 joules.

Temperature ° C.

Observed values from
curve.

Calculated values from
formulae.

o

5

1-00530

1-00446

10

1-00230

1-00206

15

1-00030

1-00024

20

1-99895

0-99894

25

0-99806

0-99807

30

0-99759

0-99757

35

0-99735

0-99735

40

0-99735

0-99735

45

0-99760

0-99757

50

0-99800

0-99807

55

0-99850

0-99894

60

0-99910

0-99910

65

0-99970

0-99972

70

1-00035

1-00036

75

1-00100

1-00100

80

1-00166

1-00166

85

1-00237

1-00233

90

1-00305

1-00301

95

1-00370

1-00370

Mean value . .

1-00012

BETWEEN THE FREEZING AND BOILING-POINTS.

253

Temperature Coefficient of the Radiation Loss.

It is not possible to obtain a very accurate measure of the temperature coefficient
of the radiation correction from the present experiments. At the same time we may,
from the different series covering different ranges of temperature, form some idea.
During a series of experiments the radiation loss remained exceedingly steady, except
that repeatedly after the calorimeter had returned from a high point the heat-loss
was found to have been increased, but tended to return to its old value with lapse of
time. On account of the slowness of the change, this occurred without producing any
serious effect on the measure of the specific heat of the water. These changes were
attributed to the effect of the small trace of occluded gases and vapour left in the
glass vacuum-jacket. It is interesting in the case of Calorimeter C to trace the
gradual alteration in the heat-loss, from series to series, during the time of the
experiments.

All of the experiments made at a mean temperature of about 30° are given in the
following table : —

Date.

Temperature.

h observed.

h corrected to
30° C.

Remarks.

February 20. .

29-09

- -04944

■05008

22. .

29-11

•04937

■05000

March 2 . . .

29-21

•04965

■05021

June 8 . . .

31-40

■05334

■05235

After trial experiment at 40°.

„ 9 . . .

32-17

•05282

•05138

„ 20 .' . .

32-81

•05429

•05230

„ an experiment at 60°.

„ 22 . . .

30-54

•05364

•05326

September 4. .

29-47

•05352

•05389

,, experiments at 60° and 70°.

„ 6. .

31-22

•05518

•05432

November 27 .

29-43

•05316

•05356

„ experiments at 0°.

January 1 . .

29-13

•05299

•05361

February -24. .

32-26

•05791

•05631

,, an experiment at 86°.

The values of the heat-loss per degree rise from the experiments on October 14 and
October 18, with rises of temperature of 8° and 5° respectively, when the flat heating-
wire was used, are : —

October 14 . 2877 '05826 '05913

„ 18 . 28'52 '05870 '05975

These values show a decided increase in the heat-loss, but was due, no doubt, to
the wire being in direct contact with the glass flow-tube of the calorimeter.

In regard to the temperature coefficient of the radiation loss, this may be calculated
from the observations in any of the different series. Series II. is the most suitable,
extending at different temperatures between 22° and 60°, over the middle of the

254

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

range. The best average of the values of h given in this series is a line represented
in the form

H/, = 11/+ -000708 (h - t),

where t is the temperature corresponding to the measurement of Eb, and tx is the
temperature corresponding to the value of Eb,. From further consideration of the
changes in the value of h from the other series, this appears to represent the tem¬
perature change of the radiation not only for Calorimeter C, but for Calorimeter E, for
the two determinations between 30° and 20°.

Taking the different values of Series II., we have, on tabulating the values of the
heat-loss, both observed and calculated, and accepting the value at 22° for H/ in the
expression given above, the following values : —

Temperature.

H observed.

H calculated.

22-16

•04619

•04619

31-40

•05334

•05273

32-17

•05282

•05328

41-02

•05939

•05954

45-49

•06306

•06271

49-68

■06669

•06562

54-61

•06867

•06916

59-80

•07220

•07285

32-81

•05429

•05373

30-54

•05364

•05212

The values at 50° and 55° are not very consistent, but it will be remembered that
the measurements at these points are not so trustworthy owing to the variation in
the experimental conditions.

On returning to 30°, as seen by the last two readings, the value of k has increased
in both cases. These two values were obtained with a rise of temperature of 1 1 ° and
5° respectively.

In regarding these large variations in the heat-loss from time to time, it must be
again emphasised that the value of the specific heat of water, owing to the method of
treatment, in no way depends on the absolute value, but only on the constancy
throughout the period of an experiment.

To prove that this was so, the order of one of the experiments in Series VIII. at
the higher points was reversed, and instead of taking the observations for the large
flow first, as was followed for all the other experiments in this series, the observations
for the small flow were obtained before those for the large flow. By this, any gradual
change in the heat-loss during the time of the experiment would have produced an
effect on the value of d in an opposite direction to the values given by the other
experiments, and would have produced twice the error.

BETWEEN THE FREEZING AND BOILING-POINTS.

255

For Calorimeter E we have the two values at 29°’92 C. and 20°'92, which are

‘05994 and ’05414.

These give for the coefficient of t in the radiation expression, the value '000645 ; or,
applying the first formula, the value of the radiation loss at 29°’92 from the value at
20o,92 = ’06051. This is within 6 parts in 10,000, and is comparable in size with
the variations from the calculated values for calorimeter C. Doubtless there would
be slight differences in the temperature coefficient of the radiation loss for different
calorimeters with different degrees of vacuum.

In Series VI., for Calorimeter C, the decrease in the radiation loss takes place with
decrease in temperature well in agreement with the other series until the experiments
at 0°, when the value of the heat-loss is increased by nearly 3 parts in 1000. The
two experiments at 1°’35 and 2°’68, both with the inflowing water at 0°T5 C., agree
however very closely with the formula as regards the temperature change in h. The
explanation of the apparent increase at these points is not altogether clear, hut may
be looked for in the very high value of the specific heat of water in the neighbourhood
of 0°, which would influence the validity of the method adopted of eliminating the
heat-loss from the large and small flows. A similar increase, although much smaller,
was noticed in the heat-loss for the same calorimeter at 4°, in Series I. Owing to the
small conduction effect at the inflow end of the calorimeter, the water in the large and
small flows enters the flow-tube, where it is heated by the electric current, necessarily
at a slightly different temperature, as was pointed out before.

Whereas this would produce no error at a part of the range where the value of d

t

was not changing rapidly with the temperature, at the freezing-point, where a very
small difference in temperature produces a large change in the value of d, it cannot be
regarded as equal in the difference equations for the two flows for the same value
of dd. Taking this into consideration, I have calculated the value of d, for the two
experiments under consideration, by extrapolating for the value of the heat-loss from
the curve for the other observations in the same series between 20° and 8°. By this
means, the value of d for each flow in the same experiment differs nearly 1 part in
1000 in the extreme case. The following are the values so obtained : —

Date.

Mean temperature.

cl large flow.

cl small flow.

November 18 ...

o

1-35

+ -0066

+ -0073

„ 22 . . .

2-68

+ -0051

+ -0060

5) ...

0-67

+ ’0072

The mean value for each experiment is larger than the value calculated in the usual
way, but for the same value of the flow the values of d are very consistent for the

256

DR, H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

different experiments, and all give identically the same temperature coefficient of
variation. For the ice experiment on March 24, of Series I., the value of h is very
nearly in agreement with the extrapolated value from Series I. and II. Hence the
above method of treatment for this experiment would give an almost identical value
of cl to the one obtained by eliminating the heat-loss from the two flows.

It is important to notice that the value of cl in this experiment, obtained from the
two flows, is more in agreement with the mean value of cl for the other ice experi¬
ments, obtained by extrapolating from the values of h in Series VI. than for the
values obtained by eliminating the heat-loss in the usual way. This points to the
fact that the values given above are more nearly correct than the values given in the
tables for the same experiments. If this be so, the indication is, that the value of the
specific heat of water rapidly approaches an exceedingly high value at 0°, and in a
remarkable way substantiates the suggestions made by Howland in his memoir in
regard to this. Further investigation is needed, however, in the neighbourhood of
the freezing-point of water, before we can say that the specific heat of water
approaches an infinite value as that point is reached. Such questions, as the con¬
tinuity of the curve for under-cooled water, render the idea quite unthinkable at
present. In view of this uncertainty, I have adhered to the lowest of the values of
the specific heat given by these measurements, and have consequently included them
both in the tables and plot. Even in this case, the change of specific heat with
temperature is very rapid, and no effect is shown by the observations taken below 4C
which would indicate a change at the point of maximum density. This, however, is
not surprising when it is considered that the point of minimum specific heat in no
way corresponds to the density curve for water.

Unfortunately, only one complete set of observations could be obtained with
Calorimeter D, with the device for getting rid of stream-line motion, owing to a crack
which, shortly after, started in the fine flow-tube inside the vacuum-jacket, and
admitted water into the jacket. This calorimeter is of sjiecial interest, as the vacuum-
jacket was supplied with a quantity of phosphorus pentoxide. The value of the
heat-loss is larger than for any of the other calorimeters, including calorimeter A,
which we exhausted ourselves to a vacuum of at least '002 millim. of mercury. This

*J

indicated that the Po05, instead of improving the vacuum as we at first thought, was
really a disadvantage. The values of the heat-loss for the four calorimeters included
in these measurements are, at 30° C. : —

Calorimeter A . 1'8 millim. flow-tube . . . '0509 watt.

„ C .... 2 „ „ ... '0500

D .... 2'8 „ „ ... -0726

„ E . . . . 3'1 „ „ ... '0600

33

BETWEEN THE FREEZING AND BOILING-POINTS.

257

Sec. 9. — Relation of the Present Measurements to the Work of other Observers.

It will hardly be necessary for me to enter into a lengthy discussion of the work of
other observers, more especially as it has been already carefully done in the original
memoirs of Rowland,* * * § Griffiths,! and Schuster and Gannon. J Since the publi¬
cation of these papers, however, a very elaborate and exhaustive series of experiments
has been made by Reynolds and Moorby§ to determine, by a direct mechanical
method, using a Reynolds brake and a steam-engine, the energy required to raise
water from a temperature slightly above . freezing to the boiling-point. The value of
the mean mechanical equivalent which they obtained is entitled to a great deal of
weight, from the minute accuracy of their measurements and the careful discussion of
possible sources of error.

It is fortunately possible, by means of the present series of experiments, on account
of their great range, to connect the experiments of Reynolds and Moorby with the
experiments of Rowland, also by the direct mechanical method, which extends
between 6° and 36° C. The absolute value of the mean mechanical equivalent
obtained by Reynolds and Moorby is 4 T 8320 joules, which is obviously less than
the same mean value obtained in the present experiments (i.e., 4*18876 joules) by as
much as 0T32 per cent.

This discrepancy in the two results may be caused by an error in the present
measurements at the extremities of the range, due to the neglecting of some
correction factor which would cause the variation curve to increase less rapidly than
it does ; blit it is far more probably due to an error in the value of one of the
constants for the determination of the electrical or heat energy. Of this latter
possibility the value of the Clark cell is still in doubt, although the value of the ohm
is fairly well fixed in absolute measure, as defined in the ‘ British Association Report’
of 1892. All of the thermal measurements are expressed in our two results to the
same scale, so that the error resolves itself into an error in the E.M.F. of the Clark
cell, which, as it enters into the equation for the determination of the electrical
energy to the second power, has twice the effect. This has been already pointed out
under the Section devoted to the Clark cell, where it was shown that if all the error
between the value of the mean mechanical equivalent obtained by the direct
mechanical method and the value obtained by the electrical method (assuming the
Clark cell equal to 1*43420 volt and the international ohm equal to 1*01358 B. A.
units) could be attributed to the Clark cell, the value 1*43420 would have to be

* ‘ Proc. Amer. Acad.,’ vol. 15, p. 75 (1879).

f ‘ Phil. Trans.,’ A, vol. 184, p. 361 (1893).

+ ‘Phil. Trans.,’ A, vol. 186, p. 415 (1895).

§ ‘Phil. Trans.,’ A, vol. 190, p. 300 (1898).

2 L

VOL. CXCIX. — A.

258

DR. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER.

reduced to 1 ’43325 volt at 15° C. Such a reduction is necessary to bring my
measurements into absolute agreement with Reynolds and MoorbyCs result. This
reduced value of the Clark cell is so nearly identical with the later absolute
dynamometer measurements as to give a most remarkable, if not coincident,
agreement between the electrical and mechanical units.

If we compare the value of the mean mechanical equivalent obtained by integrating
the values obtained by Rowland between 6° and 36°, which have been recently
corrected to the Paris Scale by a comparison of Rowland’s thermometers with the
Paris Scale, with the integrated value over the same range from the present
experiments, we find the difference between, Rowland’s value, 4T834 joules, and my
value, 4T872, in terms of the Clark cell value, 1 ’43420 volt, equal to ’091 per cent.
This is a difference of only 1 part in 2000, as deduced from the comparison of the
complete curve with Reynolds and Moorby’s result, a discrepancy which, if not
within the limits of error of our several determinations, is relatively small considering
the great range covered by these experiments. The reduced value of the Clark cell
according to Rowland would be 1 ’43355 volt, which differs from the value according
to Reynolds and Moorby by only ’3 millivolt. Owing to the slight difference in the
temperature coefficient of the specific heat between Rowland’s values and my own,
the agreement of our absolute values at any one temperature will be different at
different temperatures. At 25° our measurements, when expressing mine in terms of
Reynolds and Moorby’s, are almost exactly coincident; at 13° my value is lower
than Rowland’s by 1 part in 1000, but at 6° we are in agreement again.

Of the other direct mechanical determinations which have been made recently, we
have the work of Miculescu* in 1892, which is deserving of some mention.
Although his work is by no means above criticism, as was clearly pointed out by
Schuster and Gannon in their paper, it is of interest as showing the kind of error
which may occur between measurements by the direct method, which may be at the
same time very carefully and accurately carried out. His value, which appears to be
a mean value between 10° and 13°, is 4T857 joules. Rowland’s value at the same
temperature, about 11° C., is 4T94, while my own in terms of Reylnolds and
Moorby’s value is 4T903 joules, which, although less than Rowland’s value, is
larger than Miculescu’s.

Perhaps the most difficult part of the comparison of the present experiments with
the work of other observers is in relation to the results obtained by the electrical
method used by Griffiths and Schuster and Gannon. It is at once apparent from
fig. 17 (p. 249) that my values are widely different to the values obtained by both
these investigators, although expressed in the same values of the units used. The
explanation might at once be looked for in an error in either my Clark cells or
resistance standard ; but if it is attributed to the Clark cells used in the present
work, then the several sets of cells made at different times and from different

* ‘ Ann. de Chimie,’ vol. 27 (1892).

259

BETWEEN THE FREEZING AND BOILING-POINTS.

materials must all have involved the same error, always in the same direction. At
the same time my cadmium cells must also have been in error to exactly the same
amount and in the same direction, in order to give a ratio to my Clark cells identical
with that obtained for the cells at the Reichsanstalt, which have been compared
directly with the Cavendish standards used by Griffiths. If the error is attributed
to the value of my resistance, then we must reject the signed certificates of
II standard ohms from the Electrical Standards Committee of the British Asso¬
ciation, as well as a true ohm from the German Reichsanstalt, as being in error. It
is far more likely that the values of my constants agreed to 1 in 10,000 with those
used by Griffiths and by Schuster and Gannon respectively, and that the
difference in our results is to be attributed to some constant source of error as yet
undiscovered in our methods of calorimetry. However, the values obtained by these
observers using the same method differ by nearly 1 part in 1000 from each other,
which is not so good an agreement as exists between the measurements of Reynolds
and Moorby, Rowland, and myself, using widely different methods. At the same
time the method used by Rowland is essentially the same as that used by
Griffiths, and is subject to similar calorimetric errors. Owing to the great care
and trouble taken by Griffiths to carry out his experiments, it is difficult to see
where the difference between our two results can be. Moreover, the temperature
coefficient obtained by Griffiths, although a linear one over the range of his
experiments, is almost exactly a mean to the curve in my experiments over the same
range.

The individual observations by the present method agree very well amongst
themselves,- but although it may be correctly said that the mere repetition of
observations does not necessarily eliminate errors of experiment, yet it is possible to
vary the conditions so thoroughly by the continuous flow method of calorimetry as to
leave little room for any systematic error. In addition to varying rise of tem¬
perature, water flow and electric current, the present measurements have been made
to the same order of accuracy by varying the shape and resistance of the electric
heating conductor, by using flowT-tubes of different sizes, and by employing calori¬
meters with different values of heat -loss, this last being identical to the cooling
correction in the older methods of calorimetry.

It may be questioned whether the separate determination of the cooling effect by
special experiment and its subsequent application as a correction to calorimetric
experiments, can be relied on to an accuracy greater than 1 part in 1000. The
variations in the radiation loss measured from time to time in the present experiments
are so large that unless it had been separately determined and eliminated from the
final result for each experiment, large errors would have been introduced. Indeed, it
appears that the cooling correction is a far more uncertain factor in methods of
calorimetry than has been hitherto sufficiently realized. All questions, however,
relating to the absolute values of the standards used in the present results in no way

2 L 2

DE. H. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

2(30

affect the accuracy of the relative results, as regards the variation of the sj:)ecific heat
of water.

It is interesting to compare the absolute value of the Clark cell obtained by
assuming Griffiths’ absolute value of the mechanical equivalent at 15°, and to
express my mean value in terms of his experiments. By so doing the absolute
value becomes 4 '1975 joules, which differs from Reynolds and Mooeby’s value by
•34 per cent, or, assuming the error to be due to the Clark cell, equal to ‘17 per cent,
on D4342, which would reduce this value to 1’4318 volt at 15°. This, however, even
referred to the lowest of the latest absolute determinations, seems to be too low
a value, by as much as l millivolt, to be reconciled with the most probable true value
of the Clark cell.

It might be thought advisable, in view of the uncertainty in the electrical units, to
accept Rowland’s corrected values and express the present series of experiments in
terms of his results, which would give a mean value quite sufficiently in accord with
Reynolds and Mooeby’s mean determination. This could be done either from the
integrated value over the range of his experiments, which would tend to eliminate
errors in his method at the two extremes of the range, or by accepting his absolute
value at a temperature where he could obtain the most accurate measurement. The
present experiments over the range between 4° and 60° have already been published
(‘ B.x4_. Report,’ 1899), and were referred to Rowland’s absolute measurement at
20° C., but I think that the uncertainty in the thermometric standards used by
Rowland at that time do not warrant an accuracy greater than 1 part in about
2000, and that the mean result over the complete range of temperature referred to
Reynolds and Mooeby’s determination is more near the truth.

The value of the mean specific heat between 0° and 100° C., 4-232 joules, obtained
by Dieteeici (‘ Wied. Ann.,’ vol. 33, p. 417, 1888) in terms of the electrical units, is
obviously too large to be accounted for by an error in the electrical units, or to be
reconciled with the direct determination of Reynolds and Mooeby. The curve
obtained by Baetoli and Steacciati (‘ Beiblatter,’ vol. 15, p. 761, 1891) for the
variation of the specific heat of water between 0° and 30° by the method of mixtures
in terms of a thermal unit at 15° C. passes through a minimum point at 20° C., above
which it shows a far too rapid increase in the specific heat to be reconciled with
measurements extending as far as 100° C., unless the values pass through a maximum
point.

In 1895, Ludin (Dissert. Zurich and ‘Beiblatter,’ 1897) determined the variation
of the specific heat between 0° and 100° by the method of mixtures and showed
a minimum point at 25°, but also a maximum point at about 80°. His results are in
good agreement with the present series of experiments over a range 0° to 25 , as
sho wn in fig. 17 (p. 249), where I have plotted them in terms of a mean unit between
0° and 100° C. The excessively low minimum point shown by Baetoli and
Steacciati and by Ludin respectively, both using similar methods, suggests a

BETWEEN THE FREEZING AND BOILING-POINTS.

261

source of error common to the two. The limitation of the method of mixtures is,
however, too well known to give the complete variation curve to any degree of accuracy.

I have arranged in the following table the absolute values of the specific heat of
water every 5 degrees between 0° and 100° from my measurements for a value of
the Clark cell equal to I '43325 int. volts, and assuming the true ohm as correct,
which gives the values in terms of the mechanical units in Reynolds and Moorby’s
experiments. For comparison, I also include the measurements of Rowland and
Miculescu, and those of Griffiths and of Schuster and Gannon, to the same value
of the Clark cell.

The minimum point of the specific heat, which Rowland found at 30°, really
occurs at about 3 7° '5, but this was considered as likely by Rowland, for he says in
his memoir (p. 199), “The point of minimum cannot be said to be known, though
I have placed it provisionally between 30° and 35° C., but it may vary much from
that.” And in another place (p. 200) he says, “ There may be an error of a small
amount at that point (30J) in the direction of making the mechanical equivalent too
great, and the specific heat may keep on decreasing to even 40°.”

Absolute Value of the Thermal Capacity of Water in Joules per Calorie for
Different Temperatures between the Freezing and Boiling-points, expressed in
terms of a Clark Cell Value 1 '43325 international volts at 15° C., and the Value
of the true ohm 1 '01 358 B.A. Units.

Temperature.

s

Barnes.

Rowland.

Miculescu.

Griffiths.

i

Schuster
and Gannon.

° c.

5

4-2050

4-206

10

4-1924

4-196

4-1857

—

—

15

4-1840

4-188

—

4-1927

—

20

4-1783

4-181

—

4-1871

4-1874

25

4-1746

4-176

—

4-1816

—

30

4-1725

4-174

—

—

—

35

4-1718

4-175

—

—

—

40

4-1718

—

—

—

—

45

4-1727

—

—

—

—

50

4-1743

—

—

—

■ —

55

4-1764

—

- -

—

—

60

4-1790

—

—

—

■ —

65

4-1815

—

—

—

- —

70

4-1843

—

—

—

—

75

4-1870

—

—

—

—

80

4-1899

—

—

—

85

4-1927

—

—

—

—

90

4-1955

—

—

—

—

95

4-1983

—

—

—

—

Mean . . .

4-18326

Reynolds and Moorby’s value . . .

4-18320.

262

DR. II. T. BARNES ON THE CAPACITY FOR HEAT OF WATER

Rowland’s values are those given by W. S. Day (‘ Physical Review/ vol. 7, p. 193,
1898), corrected to the Paris scale. Griffiths’ values are those quoted by Schuster
and Gannon in their paper. At 20° and 25° Griffiths’ own temperature coefficient
is used. Schuster and Gannon’s value is given in their paper at a temperature of
1 9°*1 C. I have reduced it to 20° by the temperature coefficient obtained in my
experiments, which is very similar to Griffiths’ over a short range. It will be seen
that the values of Griffiths and Schuster and Gannon are brought into closer
agreement when corrected to the same value of the Clark cell.

Extrapolating for the values of J above 100° C. we obtain from the formula

J, = J55 (1 + -000120 (t - 55°) -b -00000025 (t - 550)2)
the following values

Temperature

Centigrade.

J.

(J55 = 4-1819).

J.

(J55 = 4-1764).

o

110

4-2127

4-2072

120

4-2190

4-2135

130

4-2255

4-2199

140

4-2321

4-2265

150

4-2390

4-2334

160

4-2461

4-2405

170

4-2534

4-2479

180

4-2610

4-2554

190

4-2687

4-2631

200

4-2767

4-2711

220

4-2931

4-2875

A glance at the complete curve for the variation of the specific heat of water with
temperature reveals at once a most interesting relation. Why should the values
drop so rapidly from the freezing point and at 37°'5 the complete character of the
curve change ? There is no discontinuous or sudden change occurring at this point
that is indicated either in the outward physical state or in the density of the water,
nor do we see any connection between the curious anomaly in the density curve at
4° C. and the specific heat at that point. It is evident we have to do here with a
new, and as yet unexplained, phenomenon.

The ideas advanced by Rowland in this connection are not, it seems to me,
altogether correct when he says : — “ However remarkable the fact may be, being the
first instance of the decrease of the specific heat with rise of temperature, it is no
more remarkable than the contraction of water to 4°. Indeed, in both cases the
water hardly seems to have recovered from freezing. The specific heat of melting-
ice is infinite. Why is it necessary that the sjDecific heat should instantly fall, and
then recover as the temperature rises ? Is it not more natural to suppose that it
continues to fall even after the ice is melted, and then to rise again as the specific

BETWEEN THE FREEZING AND BOILING-POINTS.

263

heat approaches infinity at the boiling-point ? And of all the bodies which we
should select as probably exhibiting this property, water is certainly the first.”

The identification of latent heat and specific heat which Rowland makes when he
says “ the specific heat approaches infinity at the boiling-point ” and that ;£ the
specific heat of melting ice is infinite ” is hardly tenable. Moreover, the character
of the curve as the boiling-point is reached shows no indication of approaching an
infinite value, and is entirely independent of the pressure which determines the
boiling-point. The idea of an infinite value of the specific heat at 0° can hardly be
reconciled with the idea of the continuity of the curve for under-cooled water. It is
highly probable that the specific heat approaches an exceedingly high, but
measurable value, as the freezing point is reached, and that the character of the
curve below the minimum point indicates an entirely different physical state of the
water to that above. The law governing the variation of the specific heat with
temperature above 37° '5 is directly in accord with what knowledge we already
possess of other substances, and of what our preconceived ideas might lead us to
expect.

We can draw no analogies from other liquids, since our knowledge, with the
exception, perhaps, of mercury, is now only exceedingly meagre. As the
temperature is reduced below 37°'5, may it not be that the water commences to
anticipate the formation of the solid phase, even before 0° is reached, and that the
rapid increase in specific heat indicates the effort being made to resist parting with
the internal energy necessary for formation of ice, and to form a more and more close
aggregation ? If this be true, it suggests at once the same effect for other liquids.
Can we expect to find a minimum point in the specific heat curve for other liquids in
the light of the above considerations ? I can do no more than suggest such a
possibility at present.

INDEX SLIP.

Reynolds, Osborne, and Smith, J. H. — On a Throw-Testing Machine for
Reversals of Mean Stress. Phil. Trans., A, vol. 199, 1902, pp. 265-297.

Smith, J. H., and Reynolds, Osborne. — On a Throw-Testing Machine for
Reversals of Mean Stress. Phil. Trans., A, vol. 199, 1902, pp. 265-297.

Stress, repeated — Yariation of Limiting Range with periodicity of Reversals.
Reynolds, Osborne, and Smith, J. H.

Phil. Trans., A, vol. 199, 1902, pp. 265-297.

Steel, Mild and Cast; Breaking Stress for repeated Stresses of high
periodicity. Restoring effect of a period of rest.

Reynolds, Osborne, and Smith, J. H.

Phil. Trans., A, vol. 199, 1902, pp. 265-297.

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[ 265

IV. On a Th row-Testing Machine for Reversals of Mean Stress.

Big Professor Osborne Reynolds, F.R.S., and J. H. Smith, M.Sc., Wh.Sc.,
Late Fellow of Victoria University , 1851 Exhibition Scholar.

Received March. 5, — Read March 20, 1902.

PREFACE.

Although this research is a joint undertaking, I wish to point out that except for
the idea and general design of the apparatus all the work in carrying out the design
has been done by Mr. Smith, who has made all the calculations and superintended
the execution of all the work which he has not executed himself. This undertaking
occupied some two years, being not only novel but also approaching fundamental
limits which, if passed, would have wrecked the undertaking. He has also made all
the tests. Thus whatever success we have had is entirely owing to his knowledge,
skill, and perseverance. In saying this, I do not wish to imply that I have not taken
great interest in the work, for, on the contrary, I have watched it with interest and
admiration, particularly the acumen he has shown in arranging his tests and
interpreting the results, by which he has obtained evidence of two general laws
which had not hitherto been suspected, one being that under a given range of stress
the number of reversals before rupture diminishes as the frequency increases, and the
second that the hard steels will not sustain more reversals with the same range of

o

stress than the mild steels when the frequency of the reversals is great.

Owens College, OSBORNE REYNOLDS.

February 19 th, 1902.

Contents.

Page

Historical summary . 266

Objects of the research . 267

Method of applying the stress . 268

Complete description of the apparatus . 268

Determination of the stress . ..273

Modes of vibration of a specimen . 276

The specimens : preparation, annealing, fixing . 277

Method of conducting the tests . 279

Preliminary tests for mild steel . 281

On the restoring effect of a period of rest . < . 283

Relation of limiting range to periodicity of reversals . 285

Summary of results . . , 292

Conclusion . ...296

VOL. CXCIX. - A 315,

2 M

19.9.02

266

PROFESSOR OSBORNE REYNOLDS AND MR. J. H. SMITH ON A

Historical Summary.

In 1860, Sir W. Fairbairn, using a riveted girder, carried out a series of
experiments, which seem to be the first recorded experiments on Repeated Stress.

From I860 to 1870, Wohler carried out his laborious and valuable researches on
the Fatigue of Wrought Iron and Steel. From his results published in the
£ Zeitschrift fur Bauwesen,’ Berlin, the following important points may be
deduced : —

(1) That these materials (wrought iron and steel) will rupture with stresses

much below the statical breaking stress, if such stress he repeated a

sufficient number of times.

(2) That within certain limits, the range of stress, and not the maximum stress,

determines the number of reversals necessary for rupture.

(3) That as the range of stress is diminished, the number of repetitions for

rupture increases.

(4) That there is a limiting range of stress for which the number of repetitions

of stress for rupture becomes infinite.

(5) That this limiting range of stress diminishes as the maximum stress

increases.

Wohler conducted his experiments on bars of wrought iron and steel, subjecting
them to torsional stress, bending stress, equal and opposite bending stresses, and
direct tension, with repetitions ranging from 60 to 80 per minute.

In 1874, Spangenberg repeated Wohler’s experiments, using Wohler’s
machines, and obtained similar results, also published in the ‘ Zeitschrift fur
Bauwesen.’ In 1874 also, Gerber, in the ‘Zeitschrift fur Baukunde,’ Miinchen,
suggested the following formula, as representing the results of Wohler’s
experiments :

/(max) = \ A + /(Z2 - n Ay),
where / (max) = the maximum stress,

/(min) = the minimum stress,

/ = the statical breaking stress,

A = the range of stress — f (max) ±/ (min),

and n = a constant.

Accounts of other experiments and theories bearing on this subject, are given by
the following : —

Launhardt (Zeitschrift des Architecten und Ingenieur-Vereins, Hanover, 1873).

Lippold (Organ fur die Fortschritte des Eisenbahnwesens, Wiesbaden, 1879).

Professor Mohr (Der Civil-Ingenieur, Leipzig, 1881.)

THROW-TESTING MACHINE FOR REVERSALS OF MEAN STRESS. 267

Sir B. Baker in 1886 gave the results of a series of experiments on iron and steel
(Am. Soc. Mechanical Engineers ), which were obtained with machines similar to
Wohler’s, the bars being rotated in one set of tests and subjected to bending in the
other sets, the repetitions taking place 50 to 60 times per minute.

About the same time Bauschinger^ published his important memoir on the
“Variation of the Elastic Limits,” in which it is shown that when the elastic limit
in tension is raised, the elastic limit in compression is lowered, and that by subjecting
a material to a few alternations of equal stresses, the elastic limits tend towards fixed
positions, in which positions he called them the natural elastic limits. Bauschinger
then proceeded to explain the results obtained by subjecting material to repeated
stresses, by showing that the limiting range of stress coincided with the difference of
the two elastic limits.

Objects of the Research.

The present research, which was carried out in the Whitworth Engineering
Laboratory of the Owens College, was undertaken at the suggestion of Professor
Osborne B-eynolds, who proposed an investigation of “ repeated stress ” on the
following lines: — (1) The stress should be direct tension and compression; and
(2) of approximately equal amounts, such tension and compression being obtained by
means of the inertia force of an oscillatory weight ; (8) the rapidity of repetitions
should be much higher than in the experiments of Wohler, Spangenberg,
Bauschinger, and Baker, in fact, ranging as high as 2000 reversals per minute.
The importance of these points will be seen from the following considerations : —

(1) By far the greater number of experiments on “repeated stress” have been
carried out on bars subjected to bending, the ordinary formula for stress in a bent
bar being used to calculate the stress at breaking, that is, in such experiments it has
been assumed that the distribution of stress at the breaking-down point is the same
as for an elastic bar. Calculations on this assumption are not expected to give the
tensile strength of a material for an ordinary cross-breaking experiment. This difficulty
is completely overcome, and no such assumptions are necessary, when the stresses are
direct as in the present work. The (direct) stress in a bar of metal could easily be
obtained by having one extremity rigidly connected to a part of a machine having a
known periodic motion, the other extremity being attached to a known weight.

(2) The tensile stress being in all experiments nearly equal to the compressive
stress, the elastic limits would, as shown by Bauschinger, soon be changed to their
natural positions, and the range of stress for unlimited reversals would be this
natural elastic range. Tf then, the limiting range coincides with the natural range
it will be constant whatever the rate of reversals. The author considered this point
an interesting one, and it will be found that most of the tests recorded in this paper

* ‘ Mittheilungen aus clem Mech. Techn. Laboratoiium in Munchen,’ 1886.

PROFESSOR OSBORNE REYNOLDS AND MR. J. H. SMITH ON A

2G8

were carried out in such a way as to find the variation of the limiting range of stress
as the rapidity of such reversals increased. The apparatus to be described shortly
was most convenient for such enquiry, since both the speed and the oscillatory weight
could be easily adjusted.

(3) Quite apart from the point mentioned in the last paragraph , the importance of
extending the experiments to high speeds — in view of the extensive use in recent
times of high speed machinery — is too obvious to need comment.

Short specimens of small diameter (see fig. 4), had to be used throughout, other¬
wise the apparatus would have been inconveniently heavy, and for this reason any
subsequent work on the statical strength and the elastic properties of specimens
which had been subjected to repeated stress, could not be done.

Method of Applying the Stress.

A weight is supported vertically by means of the specimen to be tested, and the
upper part of the specimen receives a periodic motion in a vertical direction by means
of a crank and a connecting rod^ The inertia of this weight gives a tension at the
bottom end, and a compression at the top end of the stroke, the change from tension
to compression being gradual. The specimen and parts are guided by suitable
bearings placed in a vertical direction. The motion was made vertical in order to
reduce the friction of the bearings to a minimum. The stresses can be changed by
varying the diameter of the specimen, the load, and the speed of revolution of the
crank. In order to enable one to calculate the stresses in the specimen, the centre of
the crank shaft must be at rest, and the crank must move with uniform angular
velocity. These conditions are obtained when the crank shaft is driven by a constant
turning effort, if the moving parts of the machine are balanced, and if at the
same time the total energy of the moving parts is invariable. The apparatus
was therefore designed to satisfy these conditions as approximately as possible
(see pp. 270 to 272).

The Apparatus.

On examining the drawings (figs. 1 and 2) of the testing machine, which show the
working parts, it will be seen to consist of a cast-iron standard having two brass
bushed hearings in its upper part. In these bearings a shaft, 4 inches diameter at
the front end, and 2 inches diameter at the back end, revolves, driven by a stepped
pulley keyed to this shaft at the back part of the machine. The standard is mounted
upon a heavy cast-iron bed-plate (weight, 14 cwts.), not shown in the drawings.

The front end of the shaft is cranked, the crank pin being 1|- inch diameter,
2 inches long, and throw h inch, and a connecting rod of peculiar form is coupled to
the crank pin. One part of this connecting rod gives an oscillatory motion in a
vertical direction to the sliding pieces directly below the crank shaft, which pieces

THROW-TESTING MACHINE FOR REVERSALS OF MEAN STRESS.

269

s

Fig. 2.

270

PROFESSOR OSBORNE REYNOLDS AND MR. J. H. SMITH ON A

include the specimen to be tested. A pin in this connecting rod at the level of the
crank shaft gives, by means of a second rod, an oscillatory motion to horizontal
sliding pieces, introduced as will be seen later (p. 270) for the purpose of having the
energy of the moving parts invariable.

The two parts P and Q of the vertical sliding pieces are connected by means of the
specimen S, which is to undergo the test. The chucks H and J for holding the
specimen were chased out internally to f-inch Whitworth thread, and the specimen
was locked by means of two lock-nuts, one at each end. The specimen was prevented
from rotating by means of a key placed in the lower bearing of the vertical sliding
piece, which fitted accurately in a key way cut in the moving spindle. The lower
bearing was bushed to allow of adjustment, and a suitable locking arrangement was
provided for it.

All the working parts were well made and exceptionally strong, of mild steel,
tensile strength 24 tons per square inch ; the pins in the connecting rod were all
case-hardened and afterwards ground to fit. The greater part of the tool work was
done by the author in the College Laboratory.

Energy of the Parts.

The horizontal sliding piece was introduced in order to make the energy of the
moving parts constant. Since the vertical connecting rod is 24 times, and the
horizontal connecting rod 18 times, the throw of the crank, the motions of both
sliding pieces will be veiy approximately simple harmonic motions, and, as both these
pieces receive their motion from the same crank pin, the velocity of one will vary as
the sine, and the other as the cosine of the angular displacement of the crank. The
sum of the squares of their velocities will be constant. The kinetic energy of the
parts will thus be constant if the total mass moving in the horizontal direction is
equal to that moving in the vertical direction.

The masses of the parts were adjusted to satisfy this condition in the following
manner : — The connecting rod and the spindles were weighed in the two positions
shown in fig. 3.

Firstly, the shorter connecting w>d A was supported horizontally and the load on
the crank pin weighed ; secondly, the longer connecting rod B was supported
horizontally and the weight on the crank pin again taken. The masses of these
parts were then adjusted until the loads on the crank pin were the same in the
two cases.

The Balancing of the Machine.

Having adjusted the masses of the horizontal and vertical sliding pieces, it was
now possible to balance these parts by placing a suitable mass diametrically opposite
to the crank pin. This balance weight was made in the form of a steel eccentric D

THROW-TESTING MACHINE FOR REVERSALS OF MEAN STRESS.

271

(fig. 2), 8|- inches diameter, ’7 inch thick, and throw 1^ inches, and was keyed to
the shaft as near as possible to the crank pin.

This arrangement introduces an unbalanced couple in a plane passing through the
centre line of the crank shaft and rotating with the shaft, and to balance this couple
a smaller eccentric d (fig. 2) of diameter 5ir inches, thickness ^ inch, and throw 1 inch
was placed near to the far end of the shaft with its centre in the axial plane passing-
through the crank pin.

So far then — neglecting the obliquity of the connecting rods, which were respec¬
tively 24 and 18 times the throw of the crank— the unloaded machine was balanced,
and the kinetic energy of the parts was constant.

FIRST POSITION SECOND POSITION

In loading the machine the system of weights used was so designed that when
one cast-iron weight of 6|- pounds was added to each of the oscillating pieces
(U, Y, fig. 2), and one semicircular yg-inch steel plate was added to each side of each
eccentric balance weight (D and d ) the machine was still balanced. It was necessary
to use the two balance weights to each eccentric (one on each side) in order to keep
the plane of the unbalanced force due to each pair in a constant position, that is, in
the central plane of the eccentric perpendicular to the shaft. Fig. 2 shows the
working parts of the machine fully loaded.

To prevent any vibration from being transmitted to the building when the machine
was running unbalanced, and to hold the machine in position, the bed-plate was
supported by four spiral springs (made of y-inch steel of square section), 3 inches

272

PROFESSOR OSBORNE REYNOLDS AND MR. J. H. SMITH ON A

diameter, which were fitted into cases specially cast on the under side of the bed¬
plate for them, their other extremities being sunk into the floor of the laboratory.
The machine was mounted on the bed-plate in such a manner as to bring the vertical
line of the oscillating piece into the centre line of the bed-plate, and about 3 cwts. of
cast-iron was then bolted to the inside of the bed-plate, and its position adjusted so
as to bring the surface horizontal.

The machine was driven from the countershaft by means of a f-inch rope and large
stepped grooved pulley, 30 inches diameter, a movable pulley being used to adjust
the tension of the driving cord.

At the left side of the bed-plate there was attached a speed indicator designed and
constructed by Mr. T. Foster, Mechanical Expert in the Whitworth Engineering
Laboratory. The form of the indicator is simple, the speed being indicated by the
rise of water in a tall glass tube due to the “ centrifugal force ” produced by setting
it in rotation by means of a spindle driven by the machine. The author is indebted
to Mr. Foster, not only for this, but for many valuable suggestions and many
excellent pieces of his workmanship during the construction of the apparatus.

A central gunmetal spindle, driven from the crank shaft by a small gut band, has
attached to it four radial vanes which revolve in a cylindrical brass box filled with
water. A glass tube rising vertically is connected to the lower part of the case, and
has a scale attached which is graduated by means of a revolution-counter to measure
revolutions per minute. The case containing the water was arranged so that the
amount of water used could be accurately regulated. Coloured water was first used,
but it was found better to use pure water as the colouring matter was deposited on
the tube, and after a few weeks made the taking of readings difficult.

In addition to this, at the back end of the machine a cast-iron bracket is bolted to
the standard. This has its upper surface planed, and on it a small table having a
revolution- counter (T, fig. 2), attached to it. The table slides on the surface, being
guided so as to allow the counter to jmss in and out of gear with the end of the
crank shaft. A steady pin was used to hold the counter in its different positions.

A lead buffer is used to receive the blow from the vertical oscillatory weight when
the specimen breaks. Two cast-iron pieces F and G (fig. 5, p. 279), keep the buffer
central and are so arranged that when F is lifted G can be removed. A conical piece
of lead is inserted in the centre of the piece F, and is directly under the vertical
spindle. The pieces of lead can easily be replaced, and it was found necessary to
replace them after every three or four tests.

Method of Lubrication.

A great amount of difficult}^ was experienced in suppl}Ting the oil to the various
bearings. A very thick oil was used for the crank pin, and an ordinary machine oil
for the other parts. In the final arrangement of the apparatus, the oil was supplied

THROW-TESTING MACHINE FOR REVERSALS OF MEAN STRESS. 273

from two large glass vessels supported by brackets fixed to the wall behind the
machine. The oil was led by means of two f-inch brass pipes to the machine, one
pipe being arranged to feed into a brass cup at the extremity of the crank pin, the
other having a number of branches passing to the various bearings in the upper part
of the machine.

The brass cup supplies the oil to the crank pin by centrifugal force, by means of a
hole passing along parallel to the centre line of the shaft and then at right angles
into the crank pin bearing. The two pins in the horizontal connecting rod were
supplied with oil from two vertical pipes having cups to receive the oil at their upper
extremities, which pipes are connected together and oscillate about a pin supported
by a bracket attached to the frame of the machine. From one pipe the oil is led
directly up the centre of the pin connected to the horizontal sliding piece, and from
the other the oil is led to the same pin but passes along a pipe attached to the
connecting rod to the pin at the other end of the rod.

Sheet iron shields are placed about the revolving parts, and these collect the oil
thrown off, carry it down the vertical rod, and allow it to drain into a cup and pipe
for carrying the oil so drained to the lower pin of this rod. The bearing of the
vertical sliding piece, above the load, receives the oil drained from the upper parts of
the machine ; the bearing below the load receives the oil from passages cut in the
sliding spindle through which the oil passes on its downward course. The crank
shaft bearings and the horizontal sliding piece bearings have each a separate oil
supply pipe.

A sheet iron trough is inserted between the frame of the machine and the bed-
plate to catch the oil. The oil is taken out of the troughs, passed through a filter,
and again used. The thick oil was used for nothing but the crank pin for some time,
but, owing to the mixing of the oils in the lower trough, the oil all became gradually
of a heavy variety, so that in about a few months the same oil was used for all the
bearings.

The machine, it was found, worked well after a few months’ running, but on
changing from slow to high speeds, or vice versa, a little trouble was always
experienced owing to the bearings heating. To help to keep the crank shaft
bearings cool, — as it was these bearings which heated most easily, — a hole was
drilled right through the whole length of this shaft, and a brass junction was
specially made for the back end ; to this junction an india-rubber pipe conveying a
stream of water was connected. The water passed along the hole in the crank shaft
to the front of the machine, where, on passing out, it was received by a pipe of
larger diameter, through which it was drained away.

Determination of Stress.

If W is the weight below the specimen, It the radius of the crank, L the length
of the vertical connecting rod, w the angular velocity of the crank pin, and A the

VOL. CXCIX. — A.

N

274

PROFESSOR OSBORNE REYNOLDS AND MR. J. H. SMITH ON A

area of the specimen, then the compressive stress in the specimen at the upper end
of the stroke is ecpial to

W or R / _ R\ _ W
(j A \ L ) A 5

and the tensile stress at the lower end of the stroke to

AY or R

S' A

i +!)+?•

The range of stress is equal to the sum of these, or is equal to

2 W or R -4- gA.

The values of W, the weights of the vertical loads used below the specimen, were
determined to one hundredth of a pound, and were as- follows : —

Spindle and lock-nut

. . 6T5

one weight

. . 12-42

two weights .

. . 18-69

three ,, .

. . 24-96

four ,, .

. . 31-23

five ,,

37"50

six ,, . .

. . 43-77

The error in the determination of the stress due to the maximum error in the
estimation of these weights would not in any case exceed '3 per cent.

The throw of the crank was measured to a ten-thousandth of an inch, the value
obtained being ’5067 inch. The maximum error in this measurement would not
affect the stress by more than *1 per cent.

The areas of the specimens were determined by finding their diameters by means
of an ordinary micrometer gauge which was graduated to ten-thousandths of an inch.
Assuming that the greatest error in actual measurement would not be more than
three ten-thousandths of an inch, then the error from this cause for a specimen
\ inch diameter would not exceed ’25 per cent.

It is thus seen that the errors incurred in the estimation of AAfi R and A are
negligible.

There are three sources of errors in the estimation of the angular velocity. They

O J J

are —

(1) The variation due to fluctuation in the energy of the parts;

(2) The variation due to fluctuation of the velocity of the engine in a cycle ; and

(3) The variation due to the fluctuation of velocity over a long interval arising

from the difficulty of regulating the motive power.

THROW-TESTING MACHINE FOR REVERSALS OF MEAN STRESS.

275

(1) To determine the extent of the variation due to the first cause, the curves of
displacement for the oscillating parts were carefully drawn to a large scale, and the
harmonics of the motion found by the usual graphical method. The harmonics were,
however, very small, and as the kinetic energy of the reciprocating parts is only
about yg-th of the total kinetic energy of the rotating parts, it is evident that the
fluctuation of energy of the parts could only introduce infinitesimal fluctuations in
angular velocity.

(2) The fluctuations of velocity of the engine in a cycle, loaded as it was by a
heavy fly-wheel and rope pulley, and connected with a long line of shafting having a
large number of heavy pulleys attached, are also negligibly small. Thus the only
real difficulty in eliminating errors in the measurement of co was found in overcoming
the secular variations of velocity.

(3) In the first series of experiments, the machine was driven by a Crossley’s
oil engine of three horse-power, but the fluctuations of velocity were not small
enough, even when the engine was working with full load, for this mode of driving
to be considered satisfactory. Although the author was not content with the
results obtained under these conditions, yet, for the sake of comparison, the results
of one set of 20 tests are given in this paper.

The machine was finally driven by the low-pressure engine of the triple expansion
experimental engines. These engines are described in a paper on ‘ The Mechanical
Equivalent of Heat,’ by Professor Osborne Reynolds and W. H. Moorby, 1898.#
The secular changes of velocity were again found to be great, and it was only after a
great number of trials that the following method (suggested by Professor Reynolds)
was hit upon to reduce them to a minimum, it being the only method suitable for
this work.

The boiler was worked at 120 lbs. pressure, and the steam was throttled so as to
reach the engine at 5 lbs. per square inch. In this way, small variations of boiler
pressure were rendered less effective in causing variations of velocity.

The engines were run so as to give out approximately 20 horse-power, and drove
by means of a rope a long line of shafting from which the power was taken to the
counter shaft of the machine by means of a 2 \ -inch belt. The surplus work was
dissipated in a hydraulic brake, also described in the paper just referred to. The
brake was not loaded in the ordinary way, but was allowed to bed against an upright
or dead-stop behind the brake ; a fairly constant flow of water was supplied to the
brake, and the resistance offered by the brake was varied by regulating the quantity
of water passing out of it.

A speed indicator, similar to the one attached to the testing machine, and
previously described (see p. 272), was driven directly from the engine, and the
heights of the water columns in the two indicators were constantly watched by

* ‘ Phil. Trans.’
2 N 2

276

• PROFESSOR OSBORNE REYNOLDS AND MR. J. H. SMITH ON A

means of a telescope and mirror, so arranged as to bring the images of the two
columns next to each other. The fluctuations of velocity of the engine and the
machine could thus he easily compared, and any slipping at once detected.

It was found that, using this method of regulation, the fluctuations of velocity of
the machine and engine corresponded with one another, and that the fluctuation
could be kept within very small limits, namely, about Tth per cent. A certain
amount of experience was necessary to ensure this steady motion for a long period,
as in varying the water passing out of the brake, a little too much either one way or
the other, oscillations of speed were set up which took some time to die away. It
was also found that the reading of the two speed indicators did not correspond at
once when the machine was started after a period of rest, but that after a few
minutes’ run they settled down to corresponding positions.

The telescope and mirror were discarded after some time, but the speed indicators
were occasionally checked in each experiment. Mr. Joseph Hall, the engine
attendant in the Whitworth Engineering Laboratory, soon became quite expert in
keeping the variations of speed within surprisingly narrow limits, even when an
experiment extended over seven or eight hours without a stop.

The author often found it impossible to be in attendance the whole time occupied
by long tests, and in such cases the machinery was left in charge of Mr. Hall. The
author found that he could leave the apparatus in his charge with the utmost
confidence.

The maximum error in the determination of the stress, due to errors in the
measurement of co, is finally estimated at '3 per cent.

Modes of Vibration of a Specimen.

The specimen may vibrate during a test in three ways, longitudinally, transversely,
and torsionally, and it is important that, either the periods of the free vibration of
the specimen do not coincide with the period of any unbalanced force in the machine,
or that the vibrations are prevented from taking effect by the use of suitable guides.

The central cylindrical part of most of the specimens was half an inch long and
'25 inch diameter. The greatest load suspended from it was 43 7 7 lbs., and the
smallest 1 2 '42. Taking 30 X 106 as Young’s modulus for mild steel, the number of
longitudinal vibrations per minute was calculated and found to be between the
limits 130,000 and 50,000 approximately.

The highest speeds at which the machine was driven with the greatest and least
loads were 1,800 and 2,500 revolutions per minute respectively. It is thus evident
that the free period of the longitudinal vibration of the specimen can never coincide
with, or be any simple multiple of, the speed of the machine, and hence can never
coincide with any periodic force arising from the imperfect balancing of the moving

THROW-TESTING MACHINE FOR REVERSALS OF MEAN STRESS.

277

The number of free vibrations per minute of the specimen vibrating transversely
was calculated, considering the specimen as a bar fixed at its end, and found to be
about 500,000 per minute, so that in this case also the free vibration need not be
considered as influencing the results.

The number of vibrations per minute executed by a specimen when oscillating
torsionally was also estimated. The calculation gives for the heaviest and lightest
loads used 1200 and 2800 vibrations per minute respectively, so that it is quite
possible that the free period of torsional vibration of the specimen might coincide
with the speed of the machine.

The key attached to the vertical sliding spindle, which works in a key-slotted bush
which can be adjusted and locked, prevents such vibration above a certain amplitude
taking place, and they are damped also by the viscosity of the oil in the bearings of
the spindle. Still, it is evident that this key and keyway cannot be fitted so
accurately as to completely extinguish a twist in the specimen as it slides with the
weight.

With a view to eliminating the effect of the torsional vibrations of specimens, a
number of tests were carried out under different conditions. It was found that when
the speed corresponding to the free period of torsional vibrations was reached, a
change in the moment of inertia of the load seemed at once to eliminate the
vibrations, whilst the unlocking of the lower bearing greatly increased them, causing
the specimen to break with fewer reversals. When the speed did not correspond
with the free period, neither the change of moment of inertia nor the conditions ot
locking effected the results (see footnote, Table III., set C).

The results of these tests are given later (p. 289), and it is seen that only when the
free vibration of the specimen coincides with the speed of the machine has this
vibration any influence on the results.

Preparation of the Specimens.

The materials used in the tests, of which the results are given in this paper, were
mild steel, best cast steel, and best Lowmoor iron. By far the greater number of
tests were carried out on specimens
having dimensions given in fig. 4.

Bars f inch diameter were cut up
into short lengths of 6 inches. The
centres were marked and small holes
were drilled up these centres for each
piece, and the pieces were then square-
centred in the ordinary wray. A rough
cut was then taken over the whole length, and the cutting of the screw, over
5 inches in length, was then commenced. This part was then finished by means of a

278

PROFESSOR OSBORNE REYNOLDS AND MR. J. H. SMITH ON A

chaser to a Whitworth Standard f-inch thread. The pieces were marked off to a
standard length and the ends cut down well below the thread. The central part
was roughed out and turned carefully as the size required was approached, a template
being used for the curved ends, which were cut to an arc of a circle f inch diameter.

The specimens were next rotated between centres and ground to size by means of
a small emery wheel which rotated rapidly in the opposite direction to the specimen,
and which was moved backwards and forwards parallel to the axis of the specimen.
The amount taken off during the grinding process was usually about one-thousandth
part of an inch all over the central parallel part.

In the early part of the work the specimens were turned, ground, and finished in
the College Laboratory, but during the latter part they were prepared by Messrs.
Carters and Co., Engineers, Salford.

A number of specimens were turned and ground very roughly in order to see the
effect of any bad workmanship, but the results were found to agree almost as well as
if the specimens had been turned and ground in the careful manner above described.

In the case of the cast-steel specimens the bars were sawn up and the short
pieces were then annealed in a gas furnace before commencing the turning process.

The Annealing of the Specimens.

The finished specimens which were to be annealed were placed inside a piece of
wrought-iron piping, 6 inches diameter, and the pipe was closed at both ends by
means of two cast-iron covers. The case so formed containing the specimens was
placed so as to stand with the specimens vertical, inside a gas furnace, and heated.

The jet of hot gases was prevented from playing directly on the case by using a
cast-iron plate which was placed opposite to the jet, and the case was rotated
frequently to ensure uniform heating. The supply of gas and air to the burner
could be adjusted as required. The process of heating up to a red-heat usually took
about half an hour, during which time the specimens were occasionally examined by
moving the upper cast-iron cover ; the gas supply was then diminished so as to keep
the furnace at a constant temperature for another half-hour, after which the burner
was taken away, the passages for outlet and inlet of hot gases plugged up by pieces
of cast-iron, and the whole allowed to cool, the cooling process usually taking from
10 to 12 hours.

On taking the specimens from the annealing furnace the thin coat of oxide was
removed from the central parallel part by rubbing it with the finest emery cloth.
This coat was, as a rule, easily removed, but in a few cases the specimens were
polished in a lathe, as the skin was found to be very hard.

The diameter of the central part was next measured by means of an ordinary
micrometer gauge, and if this part was found to be slightly tapered, the diameters at
the centre and each end were measured.

THROW-TESTING MACHINE FOR REVERSALS OF MEAN STRESS.

279

Method of Fixing the Specimens.

A lock-nut was screwed on to each end of the specimen, a hardened steel ball,
h inch diameter, was inserted in the upper chuck H (fig. 5), and that end of the
specimen which contained the centre mark was screwed into
this chuck, but not screwed home. The oscillatory weight
was then next brought up and the chuck J, which contained
another hardened steel ball, was screwed on the lower end of
the specimen until the specimen bedded against the ball, the
specimen being prevented from rotating by means of a pair of
gas-tongs with whicli the short parallel part m was gripped.

The lock-nut n was screwed tight, thus fixing the specimen
to the oscillatory weight.

J c>

The weight was now supported, and the specimen screwed
up so as to bed against the steel ball in the upper chuck, the
small force necessary for this being supplied by gas-tongs,
with which, in this case, the parallel part l was gripped.

The lock-nut r was then screwed tight, thus fixing the
specimen to the chuck H.

By the above method, one was certain of getting the
specimen into the machine without straining it, whereas, had
the specimen been fixed first to the upper chuck, it is quite
possible that the material would have been subjected to
severe torsional strains in connecting up the oscillating
weight.

In the tests carried out with the apparatus driven by the
oil engine, the steel balls were not used, and they were also
discarded in the tests on wrought iron and cast steel, since
it was found that the specimens could be easily locked in the
manner described above without using the balls.

In a great many of the tests the specimens were not
prevented from rotating, for, as previously explained, it was found unnecessary to
do so, but when necessary, to prevent oscillations, the lower bearing was locked by
means of a small brass set-screw k, \ inch diameter, which was screwed so as to press
against the outside of the loose bush which formed this bearing, thus by frictional
force preventing torsional oscillations.

Method of Conducting Tests. •

The boiler fire was generally made about 8.30 in the morning ; steam was up and
the engine was started a little before 9.30. During this time the oil supply pipes of

280

PROFESSOR OSBORNE REYNOLDS AND MR. J. H. SMITH ON A

the testing machine and engine were attended to, the cocks being adjusted to give
the necessary flow for each hearing ; the large oil vessels on the wall were filled up ;
the rope was placed on a suitable step of speed pulley, and the movable pulley was
adjusted to give the requisite tension for driving; a specimen was inserted in the
manner previously described ; the counter reading was taken ; the water inlet cock
connected to the hydraulic brake was adjusted so as to give the necessary flow of
water through the brake ; and the apparatus was then ready for the carrying-out of
a test.

Having decided upon the particular speed at which the machine was to run, the
speed of the engine to obtain this was found from the tabulated results of a series of
experiments previously made ; the flow of water from the engine brake was
regulated (see p. 275) by means of the water outlet valve, attached to the spindle of
which was a long arm, which enabled one to delicately adjust the valve opening (so
as to bring the engine to the chosen speed) ; the machine was started after the
engine had been working steadily for a few minutes.

At the commencement of a test the author usually watched the speed indicator
attached to the machine ; the engine attendant watched the engine speed indicator
and adjusted the outlet valve of the brake if necessary. The speed of the machine
gradually rose; the time taken to rise to the required speed varied from 60 to 100
seconds when the machine was started first thing in the morning, but the speed was
attained in about 20 seconds when the machine had been running for a short time
before beginning the test.

It was very important to carefully watch the speed indicator columns on starting,
as the speed of the engine had to be reduced always one or two revolutions per
minute in the first few minutes of the run, which appeared to be the time necessary
to obtain steady lubrication of the bearings ; moreover, a great amount of trouble
was saved when the brake had been carefully adjusted at the commencement of the
test, for in many experiments, when tins adjustment had been made, it was found
unnecessary to touch the outlet cock for periods of 30 minutes or even longer. On
arriving at the steady speed, it was only necessary to watch the speed indicator on
the engine, since the fluctuations of the two speed indicators were found to agree, but
still the indicator connected to the machine was examined about every 10 minutes.

Throughout the test the boiler was attended to, so as to keep the pressure as
nearly constant as possible.

The time at which the machine attained the required speed was taken by means of
a watch ; and when the speed of the engine had become steady, the counter on the
testing machine was pushed home, and the number of revolutions taken for from
10 to 100 minutes, according to the length of the time taken for the test ; after one
minute’s interval, during which the counter reading was taken, the counter was
again pushed home, and so on throughout the test.

The time at which the specimen broke was taken, and the total time from

THROW-TESTING MACHINE FOR REVERSALS OF MEAN STRESS.

281

attaining full speed to breaking was deduced and the total reversals were estimated
from this and the mean speed.

In the case of very short tests the speed was found either before or directly after
the test, in which case the oscillatory weight was connected up by a specimen of
large diameter specially kept for such work, and the machine was run for 10 minutes
to the mark on the speed indicator at which the test was done.

It is thus seen that the reversals of stress to which the materials were subjected
during the rise of speed were not taken into account. This might make a slight
difference in the short tests, but could certainly have very little effect on the results
in the case of long tests, namely, those extending over a period greater than two or
three hours. To reduce this effect as much as possible, short tests were never
carried out without previously running the machine for some time at the speed at
which they were done.

If, during a test, the lubrication of any of the bearings of the engine, shafting, or
machine failed, it was at once detected, since the speed indicator column on the
engine fell and the pitch of the (unmusical) note given out by the machine was
lowered. In such cases, the machine was at once stopped and the time taken ; the
bearing was attended to, and a new start made.

In the case of very long tests, the machine was run for five or six hours each day
according to convenience. The engine could not be used for driving on Tuesday or
Wednesday afternoons, as it was employed during those times by the students for
the experimental trials, which form part of the College Laboratory course ; so that a
test which required ten or twelve hours to complete could not be conveniently carried
out without a stop which, in some cases, extended over two or three days.

It was important to find the effects of these periods of rest on the number of
reversals required for rupture, and after several preliminary tests were completed, a
series of experiments were undertaken to investigate this matter. An account of
these experiments is given later (see p. 283).

Preliminary Tests for Mild Steel.

A great number of preliminary tests were carried out during the time in which the
oil-engine was used as the source of j^ower and also on first using the steam-engine.
In some cases the oil-engine had an ordinary pulley and cord brake attached, so as to
cause the engine to work at full load, thus reducing the fluctuations of velocity,
whereas in other cases it was used to drive the machine without any brake.

The ordinary statical test for this material gave the following results -

Yield stress . 18 ‘64 tons.

Maximum stress . 25-83 , ,,

Breaking stress . 22 '09 ,,

Percentage elongation at maximum stress . . 22.

rupture . . . . . 29,

2 o

VOL, CXCIX. — -A,

282

PROFESSOR OSBORNE REYNOLDS AND MR. J. H. SMITH ON A

The results of the endurance tests given in Table I. were obtained whilst working
under the conditions mentioned above. The specimens were turned and finished in
the College Laboratory, and approximately corresponded to the final specimen shown
in fig. 3. The diameters were not quite the same for all specimens, but varied from
*21 to ’26 inch ; the specimens were not annealed. The machine was working with
the full load, 43 '77 lbs. ; the speeds used varied from 1200 to 1500 revolutions per
minute.

The fluctuation of velocity, as shown by the speed indicator attached to the
machine, usually ranged from 4 per cent, to 6 per cent. The revolutions per minute
were determined by dividing the total revolutions by the total time from the
beginning to the end of the test.

Table I. — Unannealed Mild Steel.

Oscillatory weight, 4377 lbs. Diameter of Specimens, '22 inch to '26 inch.

Number.

Maximum stress.

Minimum stress.

Ranee of stress.

Reversals for rupture.

1

18-97

- 16-36

35 • 33

2,360

2

17-94

15-48

33 • 42

2,330

3

16-43

14-17

30-60

5,960

4

15-99

13-80

29-79

10,240

5

14-82

12-90

27-72

53,200

6

14-73

12-72

27-45

39,700

7

14-19

12-19

26-38

17,200

8

13-68

11-87

25'55

89,200

9

14-09

12-15

26-24

68,900

10

13-67

11-73

25-40

65,400

11

13-36

11-39

24-75

71,400

12

13-24

11-43

24-67

97,800

13

13-08

11-36

24-44

132,000

14

12-41

10-70

23-11

251,000

15

11-97

10-30

22-27

332.000

16

11-76

10-01

21-77

396,000

17

11-72

10-01

21-73

404,000

18

11-57

9-91

21-48

710,000

19

10-42

8-84

19-26

1,930,000
(Not broken.)

20

11-12

9-45

20-57

3,920,000
(Not broken.)

On comparing these results with those of Wohler for a similar material —
although it is impossible to choose from his list one exactly the same as the steel
used here — one sees the general similarity of the results, but is struck by the
great difference between the total reversals for any given range of stress. It is easy
to see that this difference is greater as the stress range, and therefore as the speed
increases, thus suggesting that there is a relation similar to Wohler’s for every
speed.

THROW-TESTING MACHINE FOR REVERSALS OF MEAN STRESS.

283

If the mean variation of speed is taken at 5 per cent., the variation of stress
range due to this will be 10 per cent. This great variation would seem to reduce
the value of these tests, but the author introduced them mainly to enable one to
compare results got in this way with those obtained after the fluctuations of speed
had been reduced to a minimum. In some cases, this fluctuation is of no
consequence, as will be seen from the following.

On comparing Tests 18 and 20 (Table I.) we see that for a drop of '91 ton per
square inch in the range of stress, the reversals for rupture have increased from
7'1 X 105 to over 3-9 X 106. If, then, a fluctuation of range of stress of 10 per¬
cent. had been taking place for a test at about this particular range, it is evident
that only a very small fraction of the actual reversals recorded could be effective in
damaging the material. One thus sees that, as long as the rate of change of
reversals with range of stress is small, slight fluctuations of velocity will not
appreciably affect the results, whereas, when this rate is great, it is important to
keep the speed as steady as possible.

If the limiting range of stress increases as the speed diminishes, it will be more
rapidly approached with the method of lowering the range used here, than in that
used by Wohler, for the diminution of range is got by diminishing the speeds when
the specimens are of constant diameter. Hence, after a certain point, it will not be
worth while doing long tests, since these fluctuations of velocity, however small,
would render the results doubtful.

Finding that the reversals for rupture with any given range of stress are diminished
with the speed, the author decided to limit his tests more particularly to cases for
which the reversals were less than one million. In a few cases, however, specimens
have been subjected to a greater number of reversals.

Nearly the whole of one year was spent in an attempt to get more regular results
by improving the method of preparing the specimens, by annealing, by subjecting
each specimen to a number of reversals with a small range of stress, and by
diminishing the fluctuations of speed in the manner described previously ; and,
strange as it may seem, the results were not so regular in many of the final series of
tests as in those recorded in Table I. The only possible explanation is that the
material used for Experiment I. was of more uniform quality than that used
subsequently.

On the Restoring Effect of a Period of Rest.

Since in the long tests, namely, those extending over several days, the experiments
could not be conveniently carried out without stopping, and therefore allowing the
specimen to rest, it is important to find the effect of these periods of rest on the
total reversals required for breaking. With this object in view, tests were made
with a number of specimens of mild steel, some of which were broken without
stopping the machine, while others were allowed to rest for various periods after

2 o 2

•284

PROFESSOR OSBORNE REYNOLDS AND MR. J. H. SMITH ON A

having been subjected to about half the number of reversals required to break them.
The speed of the machine and the suspended load (24'96 lbs.) were kejM the same in
all cases. The results of these tests are given in Table II.

O

Table II.— B

Oscillatory weight, 24-96. Diameter of Specimens, ’265 inch to '269 inch.

Unannealed.

Number.

Revolu¬
tions per
minute.

Maximum

stress.

Minimum

stress.

Range of
stress.

Reversals.

Total

revex-sals.

1

1948

11-56

10-23

21-79

114,000

114,000

9

1949

11-35

10-06

21-41

122,000

122,000

Q

O

1947

11-53

10-21

21-74

124,000

124,000

4

1940

11-54

10-22

21-76

47,500

47,500

5

1947

11-69

10-37

22-06

58,400

—

(After 3 days)

1933

11-51

10-21

21-72

53,800

112,200

6

1935

11-37

10-08

21-45

58,100

—

(After 5 days)

1938

11-41

10-11

21-52

60,200

118,300

7

1938

11-38

10-09

21-47

58,200

—

(After 5 days)

1942

11-53

10-02

21-55

80,600

138,800

8

1937

11-49

10-17

21-66

58,200

—

(After 11 days)

1932

11-43

10-11

21-54

58,500

116,700

9

1940

11-39

10-09

21-48

58,200

—

(After 11 days)

1936

11-37

10-07

21-44

7,500

65,700

10

1944

11-58

10-26

21-84

58,300

—

(After 4 months)

1942

11-56

10-23

21-79

30,800

89,100

11

1944

11-57

10-24

21-81

58,300

—

(After 4 months)

1 950

11-63

10-31

21-94

32,100

90,400

Annealed.

Diameter of

Specimens,

‘25 inch.

12

1947

13-14

11-62

24-76

30,600

30,600*

13

1947

13-14

11-63

24-77

30,500

30,500*

14

1947

13-09

1 1 ■ 58

24-67

15,600

(After 4 months)

1938

12-98

11-46

24-44

20,250

35,850

15

1947

13-87

11-21

25-08

15,600

—

(After 4 months)

1936

16-40

13-35

29-75

12,600

28,200

No. 15 had extended during the first part of the test, and the diameter had changed

from

•2487 to -227

1.

In these experiments, two sets of specimens were prepared all from the same bar.
The material was the same as that employed in the tests recorded in Table III. The
first set consisted of 11 specimens, which were tested without annealing. The
length of each was '63 inch, and their diameters varied from '265 to ‘269 inch.

Specimens 1, 2, 3, and 4 were broken without stopping the machine. Specimens

* These two tests are also included in Table III., Set C, Nos. 37, 38.

THROW-TESTING MACHINE TOR REVERSALS OF MEAN STRESS.

■285

5 to 1 1 were put into the machine and run for 30 minutes each — about half the
length of time required to break Specimens 1, 2, and 3. Specimen 5 was carefully
put away for three days, after which the test was completed by putting it in the
machine and running it till rupture took place. The other specimens were treated in
a similar manner, Specimens 6, 7, and 8 resting for five days, Specimens 8 and 9 for
eleven days, and Specimens 10 and 11 for four months. The result for Specimen 4 is
irregular, and is therefore rejected.

The second set of specimens was treated at a later time. They were annealed
before testing. All the specimens of this set had the dimensions shown in fig. 4,
which was adopted as the standard size in all succeeding tests.

These results show that, if a specimen is allowed to rest when the test is half
completed, there is no appreciable recovery if the period of rest is for a few days
only. They suggest that if the period of rest extends over some months the
specimens may or may not recover slightly ; the extent would appear to depend on
the treatment which it has received previous to the test.

To settle definitely the restoring effect of a long period of rest, a great many more
experiments would have to be done, but as far as this work is concerned where the
specimens were seldom allowed to rest for more than two days, the effect of this rest
on the total reversals for rupture is negligible.

Relation of Limiting Range to Periodicity of Reversals.

Under this head are given the results of experiments to determine the variation of
the range of stress with speed when the number of reversals for rupture is constant,
viz., one million.

Six bars of mild steel were purchased together, and the whole of the specimens
for which the results given in Tables II. and III. were obtained were cut from these
bars. Six samples, each 18 inches long, were cut, one from each bar, and were tested
for statical breaking-stress, &c., in the Owens College Laboratory Testing Machine,
Avliich is a Buckton 100-ton machine of the Wicksteed horizontal lever type ; the
extensions were measured over 8 inches. The figures obtained in these tests were
as follow : —

Yield-stress.

Maximum

stress.

Breaking-

stress.

Percentage
elongation at
maximum
stress.

Percentage
elongation at
rupture.

Maximum .

17-44

24-70

21-08

24-6

31-5

Minimum .

16-81

22-93

19-34

23-0

29

Mean . . .

17-12

24-54

20-47

23-5

30

Three annealed specimens of the form used in the endurance tests (dimensions
according to fig. 4) were also broken in the same testing machine, and the maximum

286

PROFESSOR OSBORNE REYNOLDS AND MR. J. H. SMITH ON A

stress measured. The mean for these three tests was 25 '81 tons per square inch ;
the yield-stress and extensions could not be conveniently measured.

During the preliminary tests (described on p. 281), it had been noticed that the
reversals for rupture for any given range of stress diminished as the rapidity of
reversals increased, and the author decided to carry out a number of tests to
investigate this point. He decided to carry out six sets of tests with the standard
form of specimen (fig. 4).

In each set of tests, specimens from every one of the six bars were used ; the load
supported by the specimen was kept constant and the speed varied. The load was
changed from set to set so that there might be six tests for any given range of stress,
each carried out at a different speed. The first test of each set was, as a rule, done
at the highest speed at which the machine could be run. The range of stress was
varied for the subsequent tests in the set by reducing the speed. It was decided to
limit, in general, the experiments to tests taking not more than two million reversals
for rupture. The results of these tests are given in Table III.

Table III.

Set A.

Annealed Mild Steel.

Oscillatory weight, 12 '42 lbs.

Diameter of Specimens, ‘245 inch to '247 inch.

Number.

Revolutions per
minute.

Maximum

stress.

Minimum

stress.

Range of
stress.

Reversals for
rupture.

1

2380

10-00

S' 95

18-95

28,090

2

2306

9-43

8-43

17-86

22,100

o

o

2240

9-01

8-05

17-06

80,700

4

2191

8-59

7-66

16-25

136,000

5

2126

7-99

7-11

15-10

248,700

6

2047

7-51

6-67

14-18

416,000

hr

t

1956

6 • 81

6-02

12-83

334,000

8

1054

6 • 87

6 " 08

12-95

682,000

9

1909

6-51

5-74

12-25

1,138,000

10

1888

6-26

5-50

11-76

1,787,000
(Not broken.)

Unannealed Mild Steel.

Diameter of Specimens, '24 inch to '25o inch.

11

2492

11-83

10-65

22-48

43,000

12

2438

11-17

10-03

21-20

34,100

13

2382

10-54

9-46

20-00

66,700

14

2356

10-09

9-02

19-11

59,000

15

2346

9-91

8-87

18-78

61,000

16

2238

9-03

8-07

17-10

73,800

17

2190

8 - 33

7-44

15-77

66,000

18

2122

8-03

7-17

15-20

226,500

19

2071

7-34

6 -54

13 ■ 88

162,700

20

2015

6-72

5-96

12-68

2,025,000
(Not broken.)

287

THEOW-TESTING MACHINE FOR REVERSALS OF MEAN STRESS.

Table III. — continued.
Set B.

Annealed Mild Steel.

Oscillatory weight, 18 '69 lbs. Diameter of Specimens, '247 inch to ’249 inch.

Number.

Revolutions per
minute.

Maximum

stress.

Minimum

stress.

Range of
stress.

Reversals for
rupture.

21

2285

13-86

12-39

26-25

4,100

22

2250

13-33

11-91

25-24

6,200

23

2150

12-36

11-00

23-36

35,500

24

2170

12-29

10-96

23-25

58,900

25

2113

11-60

10-33

21-93

86,000

26

2032

10-83

9-62

20 ■ 45

124,700

27

1934

9-88

8-74

18-62

199,700

28

1860

9-30

7-20

17-50

350,000

29

1807

8-68

7 • 65

16-33

438,000

30

1790

8-77

7-71

16-48

528,000

31

1749

8-08

7-09

15-17

843,000

32

1715

7-79

6-82

14-61

1,840,000

33

1696

7 -65

6-70

14-35

5,076,000

(Not broken.)

Set C.

Annealed Mild Steel.

Oscillatory weight,

24'9G lbs. Diameter of Specimens, '248 inch to '25 inch.

34

2150

16-09

14-32

30-41

0

35

2032

14-34

12-73

27-07

13,500

36

1962

13-52

11-97

25 • 49

30,900

37

, 1947

13-14

11-62

24-76

30,600*

38

1947

13-14

11-63

24-77

30, 5 00 f

39

1903

12-62

11-13

23-75

55,250

40

1845

11-83

10-41

22 • 24

143,200

41

1758

10-89

9-56

20 • 45

348,000

42

1693

10-10

8-82

18-92

283,000

43

1682

9-88

8-64

18-52

783,200

Set D.

Annealed Mild Steel.

Oscillatory weight, 31 '23 lbs. Diameter of Specimens, '245 inch to '249 inch.

44

1920

16-43

14-52

30 ■ 95

4,400

45

1887

15-74

13-89

29-63

11,090

46

1831

15-03

13-25

28-28

14.300

47

1776

14-31

12-57

26-88

22,600

48

1729

13-31

11-66

24-97

72,700

49

1698

12-72

11-12

23-84

92,900

50

1642

11-99

10-45

22-44

149,400

51

1609

11-63

10-12

21-75

112,000

52

1589

11-14

9-66

20-80

400,300

53

1544

10-69

9-24

19-93

540,100

* Keyed bush, of lower bearing free to turn,
„ „ „ locked.

PROFESSOR OSBORNE REYNOLDS AND MR, J. IT. SMITH ON A

o

cj

88

T a ble II I. — continued,
Set E.

Annealed Mild Steel.

Oscillatory weight, 37 '50 lbs. Diameter of Specimens, Oil inch to ‘249 inch.

Number.

Revolutions per
minute.

Maximum

stress.

Minimum

stress.

Range of
stress.

Reversals for
rupture.

54

1832

18-35

16-15

34-50

•o

55

1776

17-44

15-33

32-77

2,900

5G

] 689

16-04

14-05

30-09

13,900

57

1645

14-82

12-92

27-74

20,100

58

1587

13-81

11-98

25 ■ 79

39,600

59

1583

13-78

11-95

25-73

21,400

GO

1544

13-16

1 1 • 38

24-54

46,600

G 1

1499

12-45

10-74

23-19

112,400

62

1475

1 1 • 63

10-02

21-65

298,200

63

1441

11-03

9 • 46

20-49

650,100

Set F.

Annealed Mild Steel.

Oscillatory weight, 43'77 lbs. Diameter of Specimens, 038 inch to 049 inch.

64

1617

17-63

15-37

33-00

16,140

65

1539

15-12

13-10

28-22

17,440

66

1491

14-05

12-10

26-15

20,150

67

1501

13-99

12-09

26-08

45,200

68

1449

13-26

11-42

24-68

54,400

69

1401

12-50

10-69

23-19

96,600

70

1402

12-31

10-52

22-83

273,100

71

1372

1 1 • 98

10-21

22-19

348,400

72

1345

1 1 • 43

9-73

21-16

679,000

73

1316

11-74

9-08

19-82

542,000

Unannealed Mild Steel.

Diameter of Specimens, 038 inch to 050 inch.

74

1660

17-28

15-10

32 • 38

43,000

75

1550

14-45

12-48

26 • 93

73,600

76

1494

14-44

12-44

26-88

67,200

77

1462

13-94

12-00

25 • 94

60,600

78

1356

12-57

10-70

23-27

199,400

79

1398

12-28

10-51

22-79

106,800

80

1399

12-17

10-41

22-58

233,200

81

1326

11-86

10-07

21-93

448,000

82

1305

11-41

9-64

21-05

1,141,000

These results are probably not so regular as they might have been with specimens
all cut from one bar. Die second column of Table III. gives the speed at which the
machine was run during the test ; the reversals are given in the sixth column. The
latter were determined from the mean speed, found from the readings of the revolution

THROW-TESTING MACHINE FOR REVERSALS OF MEAN STRESS.

counter over times ranging from 10 to 100 minutes, and the total time between
starting the machine and rupture.

The following tests carried out as part of Set B, Table III. , have not been included
in that table : —

Reversals.

Speed.

a.

132,700

1855

b.

127,000

1855

c .

17,800

1777

d.

51,250

1737

e.

329,000

1775

These tests were carried out after Nos. 24, 25, 26, and 27 of Set B. Owing to the
specimen used in (a) breaking in a shorter time than was expected, the test was
repeated with the result (b). The lower bearing, which had for tests (a) and (b) been
locked, was now allowed perfect freedom to rotate, and tests ( c ) and (d) were carried
out. The lower bush was now removed and a new keyway and key were made and
very accurately fitted. Then test (e) was carried out. The large plate weights on
the oscillating spindle were changed for others of smaller diameter but of the same
weight, and the remaining tests of Table III., Set B, were completed. It is evident
from the above that in the tests ( c ) and (cl) the free period of torsional oscillation of
the specimen corresponded with that of the machine (see p. 276);

In Tables IV., V., and VI. are given the corresponding results obtained for tests
for Lowmoor iron, cast-steel, and cast-iron, respectively.

The statical tests for these materials gave the following results : —

Yield stress.

Maximum

stress.

Breaking

stress.

Percentage
elongation at
maximum
stress.

Percentage
elongation at
rupture.

f Maximum .

16-46

23-58

23-08

24

29-4

Lowmoor iron < Minimum .

16-40

23-55

21-10

21

27-5

[_ Mean . . .

16-43

23-56

22-27

22-8

28-5

f Maximum .

40-20

60-80

60-80

—

5-9

Cast-steel . . < Minimum .

39 • 45

55-30

55 • 30

—

2-5

Mean . . .

39-85

58-10

58-10

—

3-8

The breaking stress for annealed specimens of the type used in the endurance tests
was 23 T tons for Lowmoor iron, and 48 tons for cast -steel. In the case of the cast-
iron used, the breaking stress was 9 '4 tons. As the specimens used were short, the
extensions were not measured.

By far the greater number of specimens broke without any appreciable change in
diameter or length. A fair number, however, had their diameters greatly increased,
VOL. CXCIX. — A. 2 P

290

PEOFESSOE OSBORNE REYNOLDS AND ME. J. H. SMITH ON A

but there does not appear to be any definite connection between these changes of
dimensions and the range of stress or speed, as far as the author has observed. In
only one test, namely, No. 15, Table II., was a change of diameter observed on
stopping the machine before rupture. The author is led to believe that the change
in dimensions occurs, if at all, just before breaking. In only one case had flaws been
observed on stopping the machine before the breaking-point was reached, namely,
No. 15, Table IV. It showed most peculiar flaws at both ends.

Table IV. — Annealed Lowmoor Iron.

Set A.

Oscillatory weight, 12’42 lbs. Diameter of Specimens, '245 inch to '249 inch.

Number.

Devolutions per
minute.

Maximum

stress.

Minimum

stress.

Range of
stress.

Eeversals for
rupture.

1

2380

10-15

9-08

19-23

33,350

2

2308

9-33

8-35

17-68

49,200

3

2298

9 • 53

8-51

18-04

36,200

4

2217

8-82

7-86

16-68

43,250

5

2122

7-99

7-10

15-09

252,500

6

2038

7-28

6 • 47

13-75

89,700

I

2034

7-31

6-50

13-81

192,300

8

19G9

6-73

5-97

12-70

399,600

9

1893

6-23

5-50

11-73

111,600

10

1890

6-21

5 • 48

11-69

1,236,000

Set B.

Oscillatory weight, 24'9f> lbs. Diameter of Specimens, '245 inch to '248 inch.

11

2217

13-03

11-53

24-56

16,270

12

2131

12-19

10-74

22-93

43,800

13

2066

11-54

10-17

21-71

63,000

14

2019

10-77

9-45

20 • 22

85,800

15

1975

10-50

9-20

19-71

413,000

1G

1917

9-76

8-52

18-24

342,000

Set C.

Oscillatory weight,

31 '23 lbs. Diameter of Specimens, '248 inch to -25 inch.

17

1 9 1 G

12-87

11-25

24-12

62,700

18

1836

11-77

10-25

22-02

80,200

19

1701

10-28

8-88

19-16

243,800

20

1630

8 • 48

8-14

17-62

760,100

Set D.

Oscillatory weight, 4377 lbs. Diameter of Specimens, '248 inch to '249 inch.

21

1486

13-81

11-90

25-71

54,000

22

1397

12-15

10-37

22-52

100,400

23

1367

11-69

9-96

21-65

299,500

T llRO W-TESTIN G MACHINE ECU REVERSALS OF MEAN STRESS.

201

Table V. — Annealed Cast Steel.

Set A.

Oscillatory weight, 12'42 lbs. Diameter of Specimens, ’245 inch to '25 inch.

1

Number.

Revolutions per
minute.

Maximum

stress.

Minimum

stress.

Range of
stress.

Reversals fox-
rupture.

1

2382

10-22

9-16

19-38

35,680

2

2313

9-35

8-38

17-73

50,900

o

O

2303

9-51

8-50

18-01

36,900

4

2226

8-66

7*74

16-40

97,000

5

2216

8-94

7-97

16-91

47,700

6

2117

7-70

6 - 85

14-55

273,000

7

2116

7-75

6-93

14-68

95,200

8

2034

7-21

6-40

13-61

472,000

9

1963

6-77

6-00

12-77

402,000

10

1892

6-29

5 - 56

11-85

1,327,000

Set B.

Oscillatory weight,

4402 lbs. Diameter of Specimens, '246 inch to '248 inch.

11

2215

13-06

11-56

24-62

37,700

12

2163

12-56

1 1 • 09

23-65

49,400

13

2076

11-46

10-07

21-53

86,700

14

2012

10-91

9-56

20 • 47

129,300

15

1972

10-37

9-08

19-45

189,300

16

' 1917

9 • 95

8-70

18-65

337,500

17

1838

9-17

7-98

17-15

687,400

Set C.

Oscillatory weight,

31 '23 lbs. Diameter of Specimens, '248 inch to '25 inch.

18

1975

13-73

1:2-05

25-78

31,200

19

1841

11-98

10-44

22 • 42

107,400

20

1694

10-10

8-72

18-82

341,600

21

1650

9-74

8-39

18-13

2,270,000

Set D.

Oscillatory weight, 4 3 ' 7 7 lbs. Diameter of Specimens, '248 inch to ‘240 inch.

22

1474

13-46

11-60

25,- 06

92,600

23

1395

12-19

10-41

22-60

157,000

24

1363

11-59

9-89

• 21-48

265,400

25

1326

11 -07

9-40

20-47

718,000

26

1303

1

10-64

9-00

19-64

918,000

2 r 2

292

PROFESSOR OSBORNE REYNOLDS AND MR. j. H. SMITH ON A

Table VI. — Cast-iron.

Oscillatory weight, 24'9G lbs. Diameter of Specimens, *345 inch to '35 inch.

Number.

Revolutions per
minute.

Maximum

stress.

Minimum

stress.

Range of
stress.

Reversals for
rupture.

1

1941

6-67

5-91

12-58

0

•7

1493

4-01

3-46

7-47

39,200

o

O

1 496

4-02

.3-47

7-49

36,600

4

1399

3-53

3-02

6 -55

6,000

5

1396

3-51

3-01

6-52

111,600

6

1350

3 • 35

2-84

6-19

0

7

1305

3-10

2-62

5-72

620.000

Summary of Results.

The results of tests for mild steel, given in Table III., are plotted in Diagram I.,
the range of stress being taken as ordinate, and the reversals for rupture as abscissa.

THROW-TESTING- MACHINE FOE REVERSALS OF MEAN STRESS.

•X

r.i

T1 ie method used was to plot all the results ol a set of tests and then draw a curve
through the best results, that is, those giving the greatest number of reversals.
These are indicated by the small circles in the diagram. To prevent confusion, points
lying much below the curves are not shown.

The results obtained by Wohler and Baker were plotted on the same scale as
Diagram I., but on a much larger sheet. It was found that none of Baker’s and
only three of Wohler’s results came on that part of the sheet included in Diagram II.
The three points corresponding to Wohler’s figures will be found on tire diagram.

The results chosen from Wohler’s experiments were those carried out on rotating
bars (steel axles) made by Messrs. Vickers, Sons & Co. The tenacity of this material
ranged from 26'3 to 2 IF 2 tons per square inch, and the percentage extension from
15*8 to 19 ’5. The material experimented on by Baker was soft steel of tensile
strength 26 '8 to 28 '6 tons, and percentage extension 28.

The results of Wohler and Baker for their materials are

given in the following

tables : —

Number of bar.

Maximum stress.

Minimum stress.

Range of stress.

Repetitions before
fracture.

Wohler’s.

63

16-3

- 16-3

32-6

51,240

64

15-3

15 -3

30-6

72,940

65

14-3

14-3

28-6

205,800

66

13-4

13-4

26-8

278,740

67

12-4

12-4

24-8

564,900

68

FI -5

11-5

23-0

3,275,860

69

10-5

10-5

21-0

8,660,000
(Not broken.)

Baker’s.

1

16-1

- 16M

32-2

40,510

9

16-1

16-1

32-2

60,200

o

15*2

F5 • 2

30-4

68,400

4

15-2

15-2

30-4

92,070

5

15 -2

15-2

30-4

107,415

6

15-2

15-2

30-4

128,650

7

15-2

15-2

30-4

155,295

8

11-6

11-6

23-2

14,876,432

The materials used by Wohler and Baker in their tests given above, do not
correspond very well with that used in the tests carried out by the author, but they
are, however, the only results which could be reasonably, used for the purposes of
comparison.

It appears from Diagram 1. that the range ol' stress for a- definite number of
reversals diminishes rapidly as the periodicity of the reversals increases. The

PROFESSOR OSBORNE REYNOLDS AND MR. J. IE SMITH ON A

294

following Table lias been deduced from Diagram L, and shows corresponding values
of the range of stress — for rupture at one million reversals — and speed.

Mild Steel.

Range of stress for
rupture

with 1 O'5 reversals.

Reversals per minute.

Ratio of range for

10-’ reversals to yield
stress.

25 (Baker)

50 to 60

24 (Wohler)

60 to 80

—

20 ■ 9

1337

1-22

20-1

1428

1-17

19-2

1516

1-12

18-1

1656

1-06

15-2

1744

■89

12-4

1917

•72

The author does not consider the number and regularity of the tests on wrought -
iron sufficient to enable him to trace the curves for the various loads used, but
wishes to point out that the results, with one exception, are similar to, and nearly in
agreement with, those obtained for mild steel. The exception is No. 15, Table IV.
Finding that the results with wrought-iron were not coming out anything like so
regular as in the case of the mild steel specimens, and as the time for the completion
of the work was limited, more attention was paid to the cast-steel specimens in the
hope (which was realised) of obtaining more uniformity.

The results of the tests for cast-steel are plotted on Diagram II. They show the
same lowering of the range of stress, for any fixed number of reversals for rupture
as the speed is increased. The material experimented upon by Wohler which
corresponds most nearly to the cast-steel used here was tool steel made by
Firth and Sons, of tensile strength 55 tons, and extension 9T per cent. The range
for one million reversals, as deduced from Wohler’s results, is 30*9 tons per square
inch.

Wohler’s results are for bars rotated and bent : —

Number of test.

Maximum stress.

Minimum stress.

Range of stress.

Number of repetitions
for rupture.

70

17-2

- 17-2

34-4

370,975

71

1 6 • 3

16-3

32-6

694,450

72

15-3

15-3

30-6

233,700

73

14-3

14-3

28 • 6

1,528,550

It should be noticed that in these results of Wohler, the rate of change of
reversals with range of stress was jiiiite at the point corresponding to one million
reversals for rupture. This shows that what is understood as the limiting range was

295

"HROW-TESTING MACHINE FOR REVERSALS OF MEAN STRESS.

not by any means nearly approached. Diagram II. shows that in these experiments
(Table Y.) the limiting range for cast-steel was approached very rapidly as the speed
was diminished.

It is almost impossible to compare the results obtained here with those given by
Baker, as the total reversals in his case were limited to less than balf-a-million.

Baker’s results for “fine drift steel,” of tensile strength 54 tons, elongation 14 per
cent., are as follows : —

Number.

Maximum stress.

Minimum stress.

Range of stress.

Reversals for rupture.

9

29-9

-29-9

59-8

5,760

10

29-1

29 0

58-2

7,560

1 1

23-9

23-9

47-8

14,600

12

23-9

23-9

47-8

16,300

13

20-8

20-8

41-6

26,100

14

22-8

22 • S

45 ■ 6

32,445

15

18-1

18-1

36-2

157,815

16

15-2

15-2

30 ■ 4

472,500

Scd.Le for Wohier's <$ Baker's results.

PROFESSOR OSBORNE REYNOLDS AND MR. J. H. SMITH ON A

29 G

The range of stress for one million reversals, which these tests of Baker seem to
point to, lies between 27 and 28 tons.

Taking the results given in Table V. along with those of \\ ohler and Baker, we

o O o

see that in the case of cast-steel there is a great lowering of the range of stress — for
rupture with a given number of reversals — as the speed is increased.

In the following table the author’s results are added to those of Wohler and
Baker in order to show the relation between the range of stress and the reversals
per minute for rupture with one million reversals : —

Cast Steel.

Range of stress for
rupture

with 10u reversals.

Reversals per minute.

Ratio of range for

10IJ reversals to yield
stress.

30-9

60 to 80 (Wohler)

27-5

50 to 60 (Baker)

—

20-1

1320

•50

18-3

1660

•46

16-8

1820

•42

13-1

1990

•33

In the case of cast-iron the range of stress for one million reversals obtained in the
author’s experiments is approximately 5 ' 5 tons at 1300 revolutions per minute.
Wohler obtained 478 tons as the range for one million reversals for bars subjected
to repeated tensions, the limits being 0 and 478 tons; if we assume that cast-iron
behaves in the same general way as wrought-iron and steel, Wohler’s limit would
have been much greater if the stress range had been between equal and opposite
limits, pointing possibly to the same lowering of the range as the speed increases.

It is, therefore, impossible, in the case of cast-iron, to say definitely whether the
range is diminished as with the other metals experimented upon.

Conclusion.

There are many points which the author would have liked to investigate, but was
unable to owing to the great amount of time which would be required. The only
satisfactory method of procedure with experiments of the kind dealt with in this
paper is to carry out a large number of tests bearing upon any particular point, in
order to eliminate the effects of irregularities or inequalities of the materials of which
the specimens are composed. It is only in this way that one can be certain of
avoiding the inclusion of anomalous results among those from which the deductions
are made.

A little time had been spent on the effect of annealing specimens after subjecting

THROW-TESTING MACHINE FOR REVERSALS OF MEAN STRESS.

297

them to a number of reversals. The author refrains from publishing the results,
which he considers not sufficient to establish anything definite, since they vary a
great deal ; but desires to mention that the effect of such annealing appeared in
general to shorten the life of the specimen and not to restore it, as is usually
supposed.

As mentioned early in this paper, complete statical tests of specimens of size shown
in fig. 3, could not be performed, since extensions could not be measured ; moreover,
as the 100-ton testing machine had to be used, the measurements were not too
delicate. However, two specimens which had been subjected to reversals, and must
have been nearly on the point of fracture, showed a distinctly greater maximum stress
than the unused specimens.

The author desires to thank Professor Osborne Reynolds, to whom he is indebted
for many suggestions and much valuable advice and criticism during the progress of
the work, and also for the facilities which he afforded for the carrying on of the
experimental work.

VOL. CXCIX. — A.

INDEX SLIP.

Ayrton, Hertha. — Hie Mechanism of the Electric Arc.

Phil. Trans., A, vol. 199, 1902, pp. 299-33G.

Arc, Electric — Mechanism of.

Ayrton, Hertha. Phil. Trans., A, vol. 199, 1902, pp. 299-336.

Back Electromotive Force, apparent, of the Electric Arc— Its probable
Cause.

Ayrton, Hertha. Phil. Trans., A, vol. 199, 1902, pp. 299-336.

Resistance, true, of the Electric Arc.

Aybton, Hertha. Phil. Trans., A, vol. 199, 1902, pp. 299-336.

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[ 299 ]

V. The Mechanism of the Electric Arc.
By (Mrs.) Hertha Ayrton.

Communicated by Professor J. Perry, F.R.S.

Received June 5, 1901 — Read June 20, 1901.

Ever since the first discovery of the Electric Arc, nearly 100 years ago, the secret
of its mechanism has been one of the most fascinating mysteries of science. To
account for its abnormally high temperature, and for the fact that a higher P.D. is
required to send a small current than a large one through it, the arc has been
endowed with unique properties, such as a back E.M.F. of many volts, and even a
negative resistance. The measurement of this resistance alone has been the object
of a large number of experiments, made under all conceivable conditions.

The object of the present paper is to see how far this peculiar behaviour of the
arc might have been logically predicted from the known conditions of its existence,
viz., that it is a gap in a circuit, furnishing its own conductor by the evaporation of
its own material ; and to show that it is quite unnecessary to invoke the aid of a
negative resistance, or even of a large back E.M.F., to account for this behaviour.

What happens on making the Gap.

The usual explanation given for the formation of a spark or flash, on opening an
electric circuit, is that it is caused by self-induction. The interesting question
therefore arises, could an arc be struck and maintained if there were no self-

induction whatever in the circuit ? I think it could. For the surfaces of all solids

are irregular, and therefore all parts of the carbons cannot be separated at the same
instant. The parts that remain in contact will still conduct the current, but the
fewer of them that remain the greater will be their resistance. The heat caused by
this resistance must, at last, be great enough to volatilise the carbon at the remaining
points of contact, and, by the time that no part of one carbon is touching any part

of the other, the small gap will be full of carbon vapour. [As the carbon points at

each junction must volatilise as soon as they are hot enough to do so, this vapour will
be given off at a constant temperature, viz., the lowest at which carbon can volatilise.
—March 23, 1902.]

(316.)

2 2

15.10.02

300

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

To explain the further formation of the arc, we must remember that when the
carbons are separated still more all the material in the gap cannot retain its high
temperature. The access of the cold air must, I consider, turn some of the vapour
into carbon mist or fog as distinct from carbon vapour , just as the steam issuing from
a kettle is turned into visible mist at a short distance from its mouth. The interior
globular portion A (fig. I ), which is purple in the image of the arc, is, I suggest,
composed of such carbon mist, while there is an indication of a space between

Fig. 1. Enlarged image of arc and carbons with positive carbon on top. A A, pimple mist,

BB, shadow, CC, green flame.

this mist and the positive carbon which is occupied, I believe, by a thin film of true
carbon vapour.

Next the dissimilar action of the poles, met with in so many electric phenomena,
begins. Instead of both poles volatilising, so that there is a thin layer of carbon
vapour over each with a mass of carbon mist between them, the positive pole alone
volatilises, while the negative appears simply to burn away.

Besides the film of vapour and the bulb of mist, other volatile materials must go
to make up the whole substance of the arc. For the surrounding air must not only
cool the carbon vapour, hut it must unite chemically with a certain thickness of the
mist, thus forming a sheath of burning gases surrounding both vapour and mist, and
even portions of the solid carbons themselves. This sheath of gases, which is of a
brilliant green colour with solid carbons, may be seen at C (fig. 1), while B, the
shadow between it and the mist, probably indicates where the two mingle. There

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

301

must be three sorts of material in the gap, therefore, marking the three stages
through which the vapour is continually passing.

1. It starts as a thin film of carbon vapour spread over the end of the positive

carbon.

2. It then changes into the mist that lies between this vapour film and the

negative carbon.

3. Finally it burns and forms a sheath of burning gases which encloses not

only the fresh vapour and mist, but also the ends of the solid carbons
themselves.

The Conducting Power of the Vapour, Mist, and Flame .

The specific resistances of true vapours have been shown to be high, therefore
I conclude that the film over the end of the positive carbon has a high resistance,
even though it be very thin. The mist, on the contrary, is composed, 1 think, of
minute solid particles of carbon, and must, therefore, I anticipate, have a lower
specific resistance. My experiments on the flame have shown, on the other hand,
that its specific resistance is so high, compared with that of the inner purple mist,
that it is relatively an insulator, a result confirming that obtained by Luggin# in
1889. The current, therefore, flows through the vapour and the mist, and practically
not at all through the sheath of burning gases.

The Production of the High Temperature at the Crater.

To explain the great production of heat at the end of the positive carbon, as well
as the sudden change of potential that is known to exist there, it has been supposed
that a back E.M.F. of some 35 to 40 volts existed at the junction of the crater and
the arc. But if, as I suggest, there be a high resisting vapour film in contact with
the crater, the current passing through this must generate much heat, and this heat
is utilised mainly in continuously forming fresh carbon vapour, at the lowest tem¬
perature at which carbon will volatilise — to be itself turned into mist, and then into
flame. Hence it seems probable that the high and constant temperature of the crater
is kept up by the current flowing, not against a back E.M.F., but through the resistance
of a thin vapour film of constant temperature lying over the surface of the crater.
In other words, it is not the crater itself that is the source of the heat of the arc, hut
a thin film of carbon vapour, at constant temperature, in intimate contact with it.

* ‘ Wien. Sitzungsberichte,’ vol. 98, p? 1, 233.

302

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

Why the End of the Positive Carbon has its Particular Shape.

As only the part of the positive carbon that is in actual contact with the vapour
film can be at the temperature of volatilisation, evaporation can only take place at
that surface, and hence I suggest that, unless the vapour film is as large as the whole
cross-section of the positive carbon, it must dig down into the carbon and leave the
surrounding parts un volatilised, i.e., the part of the positive carbon against which the
film rests must become concave. These surrounding parts, however, are heated
sufficiently by conduction from the evaporating surface and by the hot gases
surrounding them to burn away, and so there must be a race between volatilisation
of the centre portion and burning away of the edges, which must, in all cases,
determine the shape of the surface of volatilisation. When, all other things being
equal, the gap between the carbons is small, so that the end surface of each carbon is
well protected from the air, volatilisation will gain over burning and the pit may
become very deep. When, on the other hand, the gap is large, so that the air can
easily reach all parts of the carbon except that actually covered by vapour, these
parts may burn away as fast as, or even faster than the inner portion is volatilised,
and in that case the surface of volatilisation will be flat, or even slightly convex.
It is evident, therefore, that this surface cannot, from the very nature of things, help
being concave when the distance between the carbons is short, and flat or convex
when it is long. And this is true, whether the volatilisation is due solely to a large
back E.M.F., as some have supposed, or to the resistance of a thin film of carbon
vapour, as I have suggested, or partly to one and partly to the other.

When only a small bit of the end of the positive carbon is being volatilised, the
outer edge of the carbon will not be made hot enough to burn, and the tip will remain
relatively blunt. When, on the contrary, the area of volatilisation is large, the edge
of the carbon will be burnt away and a long tapering end will be formed, terminating
in the surface of volatilisation. Further, the shorter the arc, the less easily will the
heat be able to escape from between the carbons, so that the more remains in them
to produce burning, and, consequently, the longer must be the tapering part.
Experience shows these conclusions to be true.

Why the End of the Negative Carbon assumes its Particular Shape.

The negative carbon is shaped entirely by burning away ; the heat that raises it to
burning temperature being furnished partly by the mist that touches it, and partly
by radiation from the vapour film lying against the positive carbon. The part that
the mist rests on is protected by it from the action of the air, and does not, therefore,
burn away. At the same time this part must be hotter than the remainder of the
carbon, and so the portion of the carbon near it burns away more readily than the

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

303

a

rest, leaving a mist-covered tip which is longer and slenderer, because its sides are
hotter and burn away more readily, the larger the
crater and the shorter the arc.

Hence, with a small crater and a long arc, the nega¬
tive carbon remains fairly flat, as in a , (fig. 2) ; whereas,
as the crater becomes larger, its action alone shapes the
negative carbon as dotted in b, (fig. 2), and the extra
heating due to the mist combined with the protection
which the mist offers as shown in the full line. With
a short arc, on the contrary, a small crater alone would
produce an end as dotted in c, (fig. 2), while the com¬
bined effect of the crater and mist produce the end
outlined by the full line. Finally, with a large crater
and a short arc, the crater alone would j^roduce an end
as dotted in el, (fig. 2), while the crater and mist
together would shape the negative carbon as given by
the full line in d. Experience shows that the negative carbon does shape itself in
this way under the various conditions.

g. 2. Negative carbons, a, long
arc, small crater; b, long arc, large
crater ; c, short arc, small crater ;
d, short are, large crater.

Why die Area of the Crater is not Directly Proportioned, to the Current, but

Depends edso on the Length of the Arc.

Suppose that the current and the distance between the ends of the carbons have
been kept constant long enough for all the conditions of the arc to have become
steady, so that it is “ normal” and that then the resistance in the outside circuit is
suddenly diminished. At the first instant the P.D. between the carbons must be
increased, a larger current will have to flow through a vapour film of the old
dimensions, and consequently the heat developed in it per second will increase. The
temperature of the existing vapour film cannot rise, because there is no increase of
pressure, consequently it must expand, and spread over a larger area of solid carbon.
The moment the film had expanded in the slightest degree, it would begin volatilising
carbon from a part of the surface hitherto inactive, and thus a larger quantity of
vapour per second would be volatilised. At the next instant, therefore, the quantity
of carbon volatilised per second would have increased, and the resistance of the vapour
film would have become lower, and its tendency to expand would, therefore, be
diminished on both accounts. Thus, at each instant after the change of current the
volatilising surface would increase, but more and more slowly, till its area was such
that the heat developed per second in the vapour film only just sufficed, after all
losses from conduction, &c., to keep up the volatilisation. After that, the vapour
film would cease to expand, and the surface of volatilisation would have reached its
maximum area for the new current.

304

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

The vapour film, besides radiating heat in all directions from its free surface, must
lose a certain extra amount of heat all round its edges by conduction through a ring
of the solid carbon that it does not actually touch. The heat thus lost must be
subtracted from the edges of the part of the solid carbon that the vapour film
does touch, and this part will, therefore, be just below the temperature of

volatilisation, as will also the small ring of solid carbon outside the vapour film.
Suppose, for instance, that the full line in fig. 3 is the part of the positive
carbon that is in contact with the vapour film, then the inner dotted line will
enclose the area that is actually volatilising fresh carbon, and the
space between the two dotted circles will be at a temperature
just below that of volatilisation, because the conduction of heat
from the edges of the vapour film will bring the outer circle
up and the inner circle down to a temperature a little below

that of the vapour film itself. The slightly lower temperature

of the space between the dotted circles would make it perhaps
a little less brilliant than the volatilising surface, but it would still
be very much more brilliant than the remainder of the positive carbon, so that it
must form the outer circle of what we are accustomed to call the crater, viz., the
most brilliantly white part of that carbon. The area of the crater is thus rather
larger than the cross-section of the vapour film, while the actively volatilising surface
is slightly smaller.

When the carbon vapour proceeds from a given area, the cross-section of the
vapour film will be greater the more it is protected from the cold outer air by the
end of the positive carbon. If, for instance, AB, fig. 4, were the diameter of the

Fig. 3.

Fig. 4.

Positive carbons having the same area of volatilisation. C-ABD with a long are.

C'ABD' with a short are.

volatilising surface, the cross-section of the vapour film would be greater if the end
of the carbon were CD, than if it were CTT, or, since the end of the positive carbon
is thicker the longer the arc, the cross-section of the vapour jihn is greater the longer
the arc. This film will also be able to keep a larger ring of solid carbon at a
temperature just below the lowest at which volatilisation can take place, when the
end of the carbon is CD, than when it is CTV, therefore the whole space that is just
below the lowest temperature of volatilisation, i.e., that included between the dotted
circles in fig. 3, will be greater with a long arc than with a short one, when the

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

305

surface of volatilisation is the same in each case. In other words, the area of the
crater increases with the length of the arc with a given surface of volatilisation.
Now, I shall show presently that, in the normal state of the arc, the area of the
volatilising surface is directly proportional to the current, but is independent of the
length of the arc ; it follows, therefore, that with a given constant current the area
of the crater increases with the length of the arc, as I have found it to do by actual
measurement. *

The area of the crater, then, if we define it as that part of the positive carbon
that is far brighter than the rest, is not a function of the current only, as has
hitherto been affirmed. It is a function of the current, the length of the arc, and,
until the arc has become normal after any changes have occurred in the length or
the current, of the time after the change was made. The cross-section of the vapour
film, on the other hand, is proportional to the current, as we shall now see.

The Film of Vapour in Contact ivith the Positive Carbon acts like a Back E.M.F.

Let a be that area of the film that uses its heat in volatilising fresh carbon, and
let x be the part of which the heat is lost by conduction, radiation, &c. Then the
whole area of the film is a + x, and its resistance, if we consider the thickness of the

film to be constant, is

V

a + x

where p is a constant. The heat generated per second in

p . ci-

the film varies as — - — and, of this, only - is used in volatilisation.

Cl - \- X Cl -{- X

The quantity of carbon volatilised per second is, therefore, proportional to

a n\~ an A2

> or ; - — •

Cl - X Cl -\- X [Cl -f- Xy

But, therefore, since the temperature at which volatilisation is taking place is
a constant one, viz., the lowest possible, the quantity of carbon volatilised per second
must be proportional to the area of the surface from which it is volatilised, /.<?., to a.

qa = — j where q is constant,

1 (a+xf 1

That is, a + x, the area of the vapour film, is proportional to A, the current.

Again, from the above,

a + x A2
p (a + x) <[ ’

but

therefore

Cl -f- X

V

1

7

where f is the resistance of the film,
q(a + x)

J =

A2

(!)•

VOL. CXCIX. — A.

* ‘ The Electric Arc,’ p. 154.
2 R

303

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC

Or, since a fl- x is proportional to A,

k
A

where k is constant; that is, A f the P. D. used in sending the current through the
vapour film, is constant.

Hence, no back E.M.F. at the crater is necessary to account for the great fall of
potential between it and the arc, for tire film of high resistance vapour, whose
existence I have suggested, could cause the P.D. between the positive carbon and
the arc to remain constant, exactly as if this junction were the seat of a back E.M.F.

The Apparent Negative Resistance of the Arc is caused by the True Positive
Resistance diminishing more rapidly than the Current Increases.

It has been mentioned (p. 301) that the specific resistance of the green flame is so
high as to make it, to all intents and purposes, an insulator, so that nearly the whole
of the current flows through the mist. It follows, therefore, that the resistance of
an arc of given length must depend (apart from the resistance of the vapour film)
simply on the cross-section of the carbon mist, which, as it appears purple in the
image of the arc, can easily be measured. To see how this cross-section varies when
the current is increased while the length of the arc is kept constant, I have drawn,
in fig. 5, diagrams traced from actual images, after the arc had been burning long
enough with each current and length for the P.D. between the carbons to have
become quite constant, great care having been taken to trace as accurately as possible
the exact limits of the purple centre and the green outer flame.

The resistance of the carbon mist (as distinct from that of the vapour film) may be
defined, practically, as being the resistance of that portion of the mist that lies
between the parallel planes passing through the mouth of the crater and the tip of
the negative carbon.

The mean cross-section of the mist D3, given in column 3 of Table I., has been
obtained by taking the means of the squares of the three lengths AB, CD, and EF.
The next column, giving the ratio of D3 to the current A, shows that the cross-section
of the mist increases more rapidly than the current. Column 5 gives numbers pro¬
portional to the resistance of the mist, while columns 6 and 7 contain numbers
proportional to the power spent in the mist, as obtained from these experiments and
from the equation to be subsequently referred to.

The mist carries practically the whole of the current, and, since D° increases moie
rapidly than A (column 4), it follows that in the normal arc the resistance of the mist
diminishes more rapidly than the current increases. But equation (2) above shows
that the resistance of the vapour film varies inversely as the current. Hence, with

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC,

307

solid carbons, the whole resistance of the normal arc diminishes more rapidly than the
current increases, and consequently the P.D. must diminish as the current increases.
Thus, if, in the normal arc, SY be a change made in V, and SA he the corresponding

Fig. 5. Diagram of 2-millim. normal arc between solid carbons, positive 18 millims. and

negative 15 millims. in diameter.

change produced in the current, — , has a negative value, even although the resistance

of the arc is positive, simply because that resistance diminishes faster than the
current increases, and vice versd.

2 E 2

308

MRS. H. AYRTON ON THE MECHANISM OF THE EEECTRIC ARC.

Table I. — Mean Squares of Diameters of Mist with Corresponding Currents A, and
Potential Differences V, between the Carbons. Numbers proportional to
Resistances of Mist (Column 5), and Powers Expended in Mist (Column 6).
Normal Arc.

Constant Length of Arc = 2 millims.

Solid Carbons, Positive 11 millims., and Negative 9 millims., in diameter.

1.

2.

3.

4.

5.

6.

7.

A.

V.

D2.

D*

A'

1

_

D2'

A2

Id

From

Experiment.

A2

~D2'

From

Equation.

4

51-7

4-8

1-20

0-208

3-33

3-4

6

49-0

9-8

1-63

0-102

3-67

3-68

8

48-0

16-2

2-02

0-061

3-95

3-95

10

47-0

23-4

2-34

0-043

4-27

4 " 22

12

45 • 7

34-9

2-91

0-029

4-13

4-49

14

45-1

41-2

2-94

0-024

4-76

4-76

There is Noticing to show that the P.D. betiveen the Carbons Divided by the Current

is not the True Resistance of the Arc.

Fig. 6 shows that the curve connecting the values of A7D3 given in the sixth
column of Table I., with those of the current given in the first column, is a straight
line having the equation

A3/D3 = 0-136 A + 2-86.

Fig. 6. Curve connecting the power expended in the are mist with the current. Solid carbons,

11 millims. and 9 millims. in diameter. Length of arc, 2 millims.

Hence, for a normal arc of given length, the power expended in the carbon mist is

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

309

proportional to a constant plus a term which varies directly with the current.
Dividing by A2, we obtain to, the resistance of the mist,

0136 , 2-86

TO = — - +

A

A2

Combining this with f the resistance of the vapour him on p. 306, we have for the
total resistance of the normal arc an expression of the form

/ + to = — + — •

But I have shown* that an equation of the form

V = a + bl + c-^~

exactly fits not only all the numerous measurements that I have myself made ot
simultaneous values of the P.D. between the carbons, the current, and the length of
the arc, but also all the similar experiments made by other observers when both
carbons are solid. When l, the length of the arc, is constant, this equation becomes

where y and 8 are constants,
resistance of an arc,

Hence, dividing by A, we have, for the total apparent

Thus, by considering only the way in which the resistances of the vapour film and
of the carbon mist respectively must vary with the current, we arrive at an equation
for the resistance of exactly the same form as is obtained by dividing by the current
the values found experimentally for the P.D. between the carbons. So that instead
of an arc consisting of a circuit of low resistance combined with a back E.M.F., it may
well be that its apparent resistance, i.e., the ratio of V to A, is its true resistance;
or it may be that, if there is any back E.M.F. at all, it is very much smaller than has
hitherto been supposed.

Both the Resistance of the Arc and the P.D. between the Carbons depend not only on
the Current and the Length , but also on How Lately a Change has been made in
Either and on What that Change teas.

The whole resistance of the arc depends on the cross-sections of the vapour film
and the mist, and on the distance between the carbons. Now I have shown that

* ‘ The Electric Arc,’ p. 186.

310

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

when the P.I). between the carbons is changed — increased, say — th e, first result must
be an increase of current, while the second is a corresponding increase in the cross-
sections of the vapour film and the mist, causing a diminution of the resistance, and,
consequently, of the P.D. between the carbons. Thirdly, if the new current is kept
constant long enough, the end of the negative carbon burns away to a longer
slenderer point, allowing more of the mist to escape, so that it takes a smaller cross-
section, and, consequently, both the resistance and the P.D. increase again, although
they never reach such high values as they had with the smaller current.

Fig. 7 is useful as showing at a glance how the resistance and the P.D. depend
upon the time that has elapsed after a change of current. When the arc is normal,

c

Time.

Fig. 7. Suggested simultaneous time-changes of P.D., current and resistance.

in the first instance, A B, A ' B', and A" B" represent the curves connecting the P.D.,
the current, and the true resistance of the arc respectively with the time. When
the P.D. is increased from B to C, the resistance does not alter at the first instant,
but the current rises to C'. If it is then kept constant at C', the surface of volatilisa¬
tion next increases in area, the resistance falls to I)' , and the P.D. consequently
falls to D. After this the carbons begin to grow longer points ; the cross-section of
the mist diminishes, the resistance, therefore, increases to E ", and the P.D. with it
to E. The arc has now become normal again, so that the curves are all now parallel
straight lines, the current higher than before, and the P. D. and resistance lower.

Thus any alteration that is made and maintained in the arc sets up a series of
changes in its resistance and, consequently, in the P.D. between the carbons, that

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

311

cease only when the arc becomes normal again. In other words, when an arc
of given length, with a given current flowing, exists between given carbons, neither
the resistance nor the P.D. between the carbons has any fixed value, except when the
arc is, and continues to be normal. In all other cases each varies, within certain
limits, according to the time that has elapsed since either the current, or the length
was altered, and according to what change was then made.

That the P.D. does actually undergo alterations of the kind just described, after
a change of current, is evident from fig. 8, the curves in which were plotted from
actual experiments made in 1893. For these curves, the current was suddenly
altered when the arc was quite normal, and was then kept constant at the new value
while the P.D. continued to alter, the time change of P.D. being noted. The first
change of P. D. — the rise or fall that must have instantaneously accompanied the

Fig. 8. Experimental curves showing simultaneous time-changes of P.D. and current with a constant
length of arc of 1 millim. Solid carbons, 18 millims. and 15 millims. in diameter.

change of current, when the resistance outside the arc was suddenly diminished—
was too quick to detect. Indeed, it was only after assisting in carrying out these
experiments that it occurred to me that we ought to have seen it, and that, on trying,
I found I could sometimes detect it and sometimes not.*

The rapid change in the P.D. while the area of the crater and the cross-section of
the mist were altering, is very marked, however, as well as the slow rise or fall of
P.D. accompanying the diminution or increase in the cross-section of the arc due to
the change in the cross-section of the carbon ends.

Why Measurements of the Resistance of the Arc made under the same Apparent
Conditions Differ in Value and even in Sign with Different Experimenters.

The dependence of the resistance of the arc on its previous history, as well as on
the actually existent length and current, has an important bearing on the question

* It is only when the carbons are cored that it can be detected. The reason will be explained later.

312

MRS. II. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

of measuring the resistance by means of a small superimposed alternating current.
Such a method has been employed by many experimenters, but the results have
not shared the similarity of the methods ; for while von Lang and Arons found, in
1887, that the arc had a 'positive resistance, Messrs. Frith and Lodgers, in 1896,
found that it had a negative one with solid carbons.

We shall now see the reason of this disparity, and first it may be well to recall
shortly the reasoning on which the method is based..

The equation Y = E + A r may be taken to represent the connection between the
P.D. between the carbons, the current, and the length of the arc, whether it has a
variable E.M.F., a constant E.M.F., or none at all. For in the first case E will be
variable, in the second constant, and in the third zero. In any case SY/SA = r,
only when such a small quick change is made in V and A that neither E nor r is
made to vary by it.

Instead of a single small quick change of current, the experimenters superimposed
a small alternating current on the direct current of the arc, and measured the average
value of SY/SA, or its equivalent. Obviously, if the alternating current left the
resistance and any back E.M.F. that might exist in the arc unaffected, this was
a true measure of the resistance of the arc. But if the alternating current changed
both or either of these, then instead of being equal to ry we should have

if there is a back E.M.F., and if both it and the
resistance varied with the alternating current ;

if there is a back E.M.F., and if it alone varied ;

if there is no back E.M.F., or if the resistance alone
varied.

None of the experimenters, as far as I am aware, applied any but a few imperfect
tests to see whether the alternating currents they employed affected the resistance
of the arc or not, and it was, I believe, because the resistance teas affected, in every
case, that such diverse results were obtained. The low frequency of the alternations
was the probable source of error, for I shall now show that, with a given root mean
square value of the alternating current, the average value of SY/SA varies not only
in magnitude, hut even in sign, with the frequency of that current.

Effect of the Frequency of the Superimposed Alternating Current on the

Value and Sign of SY/SA.

I have shown (p. 310) that when a sudden increase of the current is made and
maintained the P.D. has three successive stages of variation. It first rises (BC, fig. 7),
then falls (CD), then rises again (DE), but not so high as before, and after this it

SY

SA

r 8E , A Sr

r+sX + AsX

hV

SA

= r +

SE

SA

SY

SA

— r + A

Sr

SA

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

313

remains constant. In dealing with a superimposed alternating current there is, of
course, no sudden increase and diminution, everything is gradual. The three changes
of P.D. do not, therefore, act separately — they overlap. At any moment, for instance,
when the current is increasing say, the increase may be considered due to the
addition of successive small increments of current, so that the P.D. has a tendency to
rise on account of the last added increment of current, to fall on account of the
diminution of resistance due to the last but one, and to rise on account of the
re-shaping of the carbons following the last but two. If the frequency of the
alternating current is very low indeed, so that the current changes very slowly, all
three of these tendencies will be in force at each moment, and the actual change
of P.D. will be the resultant of the three. If the frequency is so high that the
shapes of the carbons never change at all, but so low that the area of the volatilising
surface can alter, only the first two tendencies will be operative ; while, if the
frequency is so high that the area of the volatilising surface remains constant, the
resistance of the arc will not alter at all, the current and P.D. will increase and
diminish together and proportionately, and, unless the arc contains a variable back
E.M.F., SV/SA will measure the true resistance of the arc.

The influence of the frequency of the alternating current on the magnitude and
sign of SV/SA is traced in fig. (J. PR represents the time occupied by one complete
alternation, whatever that time maybe. If, for instance, the frequency is 50 complete
alternations per second, PR represents one-fiftieth of a second ; if the frequency is
5000, PR represents one-five thousandth of a second. PSQTR represents the time
change of current with any frequency. When the alternations are so slow that the
arc remains normal, the change of P.D., SV, for a given small change of current, WS
say, is the resultant of three such changes as BC, CD, and DE (fig. 7), and it is in
the opposite direction from the change of current. The P. D. time curve is something
like PXQYR (fig. 9) therefore, and SV/SA is the mean of such ratios as ZX/WS, and
is therefore negative.

When the frequency is raised, so that the carbons never have time to alter their
shapes completely before the current changes, the third variation, DE, (fig. 7, p. 310)
is smaller than with the normal arc, so that 8 V is greater negatively, and SV/SA
must, therefore, have a larger negative value than when the arc is normal, and such
a curve as PXjQY^t would be the P.D. time curve in this case.

When the frequency was so high that the carbons never altered their shapes at
all, but the volatilising surface underwent the maximum alteration, the third
variation (DE fig. 7) would be absent altogether, and therefore SV would undergo
the greatest change it was susceptible of in the opposite direction to the change of
current, so that PX2QY2R is then the P.D. time curve,, and SV/SA has then its
maximum negative value, and is the mean of such ratios as Z2X2/WS.

With a further increase of frequency, the area of the volatilising surface would
never have time to change completely, so that SV would be the resultant of two

vol. cxcix. — a. 2 s

314

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

such changes as BC and CF, say, (fig. 7) only ; SY/SA would therefore have a smaller
negative value than with the lower frequency last mentioned, and the curve denoting
the time change of P.D. might again be PX^YjPt, or it might be PX3QY3R, if the
frequency were high enough. When the frequency was so great that the two P.Ds.,
BO and CD (fig. 7) were exactly equal, the P.D. would not alter at all when the
current was changed, SY/SA would be zero, and the straight line PQB would be the
time change of P.D. curve. When the frequency was further increased, the change
of P.D. would be-the resultant of two such changes as BC, CG (fig. 7), the total change
of P.D. would therefore he in the same direction as the change of current, the P.D.

Fig 9. Suggested curves connecting current and P.D. with time for different frequencies of the same
small superimposed alternating current, when the direct current and length of the arc are constant.

time curve would be like PXtQY4R, and SY/SA would be + Z^Xy'WS. Finally,
when the frequency was so great that the area of the volatilising surface never
altered at all, the change of P.D. would be BC (fig. 7) alone, the P.D. time curve
would be PX-QYJt, SY would be Z-X-, and SY/SA, or Z-X-/WS would measure the
true resistance of the arc, even if there is a back E.M.F. in the arc, unless that back
E.M.F. varies with the current.

MRS. H. AYRTON ON TIIE MECHANISM OF THE ELECTRIC ARC.

315

Thus, by applying the same alternating current, but with different frequencies, to
a direct current arc, SY/SA can be made to have any value from a fairly large
negative value to the true positive value. It is easy to see, therefore, how different
experimenters might get very different values and even different signs for the
resistance of the arc, when they measured it by means of a superimposed alternating
current ; and fig. 9 shows the imperative necessity of some rigorous proof that the
alternating current has not affected the resistance of the arc before any such
measurements can be accepted as final. I shall presently show how such a proof can
be obtained, but first it will be interesting to see how, with an arc of given length,
and with a given current flowing, the value of SY/SA is connected with the frequency
of the alternating current, and what sort of frequency is required in order that the
resistance of the arc shall not be affected by. this current.

To find the Curve connecting SY/SA with the Frequency ofi the Superimposed
Alternating Current , and to see with what Frequency SY/SA Measures the True
Resistance of the Arc.

Take an arc of 2 millims. with a direct current of 10 amperes flowing. For the
arc to remain normal when the small alternating current is superimposed on it, the
frequency must be practically zero, for each alternation must take many seconds
instead of only a small fraction of a second. Now the equation I have found*
between Y, A, and /, in the normal arc with solid Apostle carbons is

Y

38-88 -f 2-07^ +

11 66 + 10-5Y
A

therefore the normal SY/SA = — ^ ^ ~|~a ^ ^ = — 0-33, when l = 2, and A = 10.

At

The first point on the curve connecting SV/SA with the frequency of the alterna¬
ting current has, therefore, the co-ordinates 0, and — 0'33 (A, fig. 10).

The value found for SY/SA by Messrs. Frith and -Rodgers]' with the same carbons,
direct current, and P.D. was about — 0 '8, more than double the normal value, which
shows that the alternating current they superimposed was making the resistance of
the arc vary to an extent that made the P.D. follow some such curve as PX2QY2R
(fig. 9). They also found that varying the frequency from 7 to 250 complete alterna¬
tions per second made no difference in the value they obtained for SY/SA. Therefore
the curve connecting SY/SA with the frequency must fall steeply from A, the point of
no frequency, to B, the point for a frequency of 7, and must be practically horizontal
from B to C (fig. 10). Hence Messrs. Frith and Rodgers’ observations cover the
portion BC of the curve.

* ‘ The Electric Arc,’ p. 184.
t ‘ Phil. Mag.,’ 1896, vol. 42, Plate 5.

316

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

The next point D, is obtained from Mr. Dubdell’s work. In his remarkable
paper* on “Rapid Variations in the Current through the Direct Current Arc,” he
said, “ I tried to record the transient rise in P.D. for the solid arc by means of an
oscillograph, the sudden increase of the current being obtained by discharging a
condenser through the arc. This experiment was successful, and a transient rise in
P.D. was observed, the P.D. and current increasing together, hut only for about
1/5000 second.” It is clear from this that SV/SA must at least begin to be positive
with a frequency of 2,500 complete alternations per second; and D where OD = 2,500
may be taken to be the point near which 8V/SA changes its sign.

To the right of D the curve must continue to rise, as indicated in fig. 10, more and
more slowly, as it approaches the horizontal line whose distance from the axis of
frequency represents the value of SV /SA which is the true resistance of the arc. The
curve must finally become asymptotic to this line, since when once a frequency is
nearly reached with which the alternating current does not practically affect the
resistance of the arc, increasing the frequency will not alter the value of SV/SA.

My equation above shows that the resistance of the particular 2-millim. 10-ampere
normal arc under discussion cannot be greater than 4 '6 3 ohms, nor less than 0'62
ohm ; for if there is no back E.M.F.,

38-88 + 2-07 x 2 11-66 + 1054 x 2

10 + 100

= 4-63

and if there is the largest possible back E.M.F., viz., 38 '88 -f

11-66

A

volts (for it is

impossible to imagine that terms involving the length of the arc can belong to a back

17' T\/T XT’ \ x. 2'07 X 2 , 10‘54 >< 2 A ™

E.M.F.), then r= 1Q + — — — = 0-62.

100

Thus the curve cannot rise higher than the horizontal line SV/SA = 4'63, and it
must rise at least as high as SV/SA = 0‘62. Consequently, as the lower curve in
fig. 10 shows, the true resistance of this particular arc could not be measured with
a superimposed alternating current having a frequency of less than at least 8000
complete alternations per second, even if there were a back E.M.F. as great as
40 volts. And if, as I have suggested, the back E.M.F. is zero, or at least very
much smaller than 40 volts, the frequency would have to be many times as high for
SV/SA to be on the horizontal part of the curve, i.e., for the alternating current not
to alter the resistance of the arc.

The Form of the P.D. Time Curve indicates whether the Resistance of the Arc is
Affected by the superimposed Alternating Current or Not.

The final test as to the frequency being high enough not to affect the resistance of
the arc must, of course, be the finding, with the same root mean square value of the
* ‘ Journal of the Institution of Electrical Engineers,’ vol. 30, p. 232.

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

317

alternating current, of two frequencies, differing by many thousands of alternations
per second, that would both give the same value of SV/SA. This would show that
the horizontal part of the SV/SA frequency curve (fig. 10) had been found.

A very good first test, however, is furnished by the curve connecting the P.D.
between the carbons with the time, for this curve is unsymmetrical with respect to
the corresponding current curve, when the resistance is affected, for the following-
reasons.

I have shown that the change in the area of the volatilising surface of the crater
that is due to any change of current follows after the change of current and requires
time for its completion. If, therefore, a superimposed alternating current is affecting
the resistance of a direct current arc, the P.D. required for any given current must

Fig. 10. Suggested curve connecting 8V/SA with, the frequency of a superimposed alternating current
of constant root mean square value, when the direct current and the length of the arc are constant.

be higher when the current is increasing than when it is diminishing. A current of
10 amperes, for instance, would require a higher P.D. when it came after 9 and
before 1 1 amperes than when it came after 1 1 and before 9 amperes, because in the
first case it would be flowing through an arc of which the cross-section had been
made by some current less than 10 amperes, and in the second by some current
greater than 10 amperes.

I have applied this test, with very satisfactory results, to some experiments in
which it is quite certain that the alternating currents must have affected the
resistances of the arcs, because they had frequencies of only 47 and 11 5- alternations
per second respectively.

The experiments formed part of a valuable series carried out in 1896 by Messrs.

318

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

Ray and Watlington, two students at the Central Technical College, in continua¬
tion of the researches of Messrs. Frith and Rodgers. The carbons were solid, and
the direct current normal arc carried a current of 10 amperes with a P.D. of 45
volts. Simultaneous values of the current and P. D. were taken with the Joubert
point by point method. Some of these are given in Table II., in which the columns
headed V,- are the P.D.’s with increasing currents, and those headed Yd with
diminishing currents ; while V, and V/ belong to the smallest and largest current
respectively.

It may be seen at a glance that in every single instance the P.D. for the same
current is higher when the current is increasing than when it is diminishing. For
instance, with the lower frequency the P.D. corresponding with a current of 11
amperes is 42 ’4 volts when the current is increasing, and only 41 ’8 volts when it is
diminishing. And with the higher frequency it is 43 ‘2 volts with increasing current,
and only 42 ‘0 with a falling current. Hence, we are supplied with a very simple
test as to whether the superimposed alternating current changes the resistance of the
arc or not. It is only necessary to take the wave form of P.D. and current by means
of an oscillograph, and to observe whether the P.D. corresponding with each current
is the same with increasing as with diminishing currents. If the two P.Ds. are
different, the resistance is being altered, if they are alike, it is not.

Table II. — Instantaneous Values of Corresponding Currents and P.Ds. with Small
Alternating Current Superimposed on Direct Current of 10 Amperes in the Arc.

P.D. with Direct Current alone, 45 volts.

Solid Apostle Carbons : Positive, 1 1 millims. ; N egative, 9 millims.

A.

V.

Frequency, 47.

vs.

Frequency, 115.

V,.

Vz.

V,,-.

A.

Vi.

V.

V.

8-425

45 • 6

8-65

4G

9-0

—

44-8

—

. 44-0

9-0

—

45 ■ 6

—

44-9

9-5

' -

44-2

—

43-3

9-5

—

45-2

—

44-2

10-0

—

43-7

—

42-8

10-0

—

44-6

—

43-5

10-5

—

43-1

—

42-1

10-5

—

43-9

—

42-9

11-0

—

42-4

—

41-8

11-0

—

43-2

—

42-0

11-5

—

42-0

—

41-4

11-25

—

—

42-2

—

12-0

—

41-7

—

41-2

—

—

—

—

—

12*25

—

—

41-4

—

- -

—

—

—

—

MRS. II. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

319

How to Ascertain with Certainty whether there is a Constant or a Variable Back
E.M.F. in the Arc or None , and how to Jincl the True Back E.M.F. if there
is One.

Returning to the equation
we have

SA — SA

V = E + A r,

SV SE , . . Sr . , ,, ,

— = — + r + A — when noth R and r vary,

X A X A oA.

and hence

TT A A SE . , Sr

v-asa=e-aIa-a'sa-

If the alternating current with which SV/SA is measured is of such high frequency
that it does not alter the resistance of the arc, and if, also, the back E.M.F. is
constant, or, being variable, the alternating current is too small to affect it, then

\ — A SY /SA = E.

To see whether the arc has any back E.M.F. at all, therefore, it is only necessary
to measure SV/SA with a superimposed alternating current of a frequency that has
been found not to affect its resistance and to subtract A SV/SA from Y. If the result
is zero, the arc has no back E. M. F. If it is not zero, S V /S A must be measured in the
same way for other arcs differing widely in current and length. If all the values of
Y — A SV/SA thus obtained are equal, or nearly so, the arc has a constant back
E.M.F. which is equal to this value. If A' — A SY/SA is not the same for all the arcs,
but varies according to some definite law, then there is a variable back E.M.F. which
may or may not be affected by the alternating current used to measure SV/SA.

Suppose, for instance, that two measurements of SY/SA were made, using the same
direct current and length of arc, but different alternating currents. If one of the
alternating currents had a root mean square value equal to one per cent, of the direct
current, and the other a value equal to five per cent., one would be five times as great
as the other, and yet both would be small compared with the direct current. It
would, of course, be possible to make the frequency of each of these currents so great
that the resistances of the arcs to which they were applied were not altered by them.
Yet it would not necessarily follow that when this had been done the two values ot
SV/SA thus obtained would be equal. For the back E.M.F. might vary not with the
frequency of the alternating current, but with its magnitude. If, therefore, it were
found that E was variable, it would be necessary to measure SY/SA with smaller and
smaller alternating currents, till two were found which, while differing considerably
from one another, both gave the same value of SY/SA. Only a value obtained in this
way could be accepted as measuring the true resistance of the arc, and Ar — A SY/SA
would be the true back E.M.F. of the same arc.

320

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

The Changes Introduced into the Resistance of the Arc by the Use of

Cored Carbons.

Next let us consider the explanation of the marked effects produced by introducing
a core into either or both of the carbons. These are of a two-fold character ; first,
those such as Professor Ayrton published at Chicago in 1893, viz. :

(1.) The P.D. between the carbons is always lower for a given current and length
of arc, when either or both of the carbons are cored, than wdien both are
solid.

(2.) With a constant length of arc and increasing current, the P.D., which
diminishes continuously when both carbons are solid, either diminishes less
when the positive is cored, or after diminishing to a minimum remains
constant over a wide range of current, or even increases again.*

(3.) It requires a larger current with the same length of arc to make the arc
hiss when the positive carbon is cored than when both are solid.

Secondly, there are the facts connected with the influence of cores on the small
change of P.D. accompanying a small change of current, to which attention was
first drawn by Messrs. Frith and Rodgers in 1 8 9 6. t These facts, which were
physically correct, although, as I have already shown, they were wrongly interpreted
at the time, are embodied in the following wider generalisations which I have deduced
from the results of my experiments, and from theoretical considerations.

(1) When, on a direct current arc, an alternating current is superimposed which
is small, but yet large enough for the resistance of the arc to be altered
by it, the average value of SV/SA is always more positive! when either
carbon is cored than when both are solid, and most positive of all when both
are cored, all other things being equal.

(2) The frequency of the alternating current that makes SV/SA begin to have

a positive value is lower when either carbon is cored than when both are
solid, and lowest when both are cored.

(3) The value of SV/SA, with a given root mean square value of the superimposed

alternating current, depends not only on the nature of the carbons and on
the frequency of that current, but also on the magnitude of the direct
current, and on the length of the arc.

* Nebel observed the fact that the P.D. fell to a minimum and then rose again, in 1886, but as he used
cored positive carbons only, he did not discover that this form of curve was peculiar to those carbons.

t “The Resistance of the Electric Arc,” ‘Phil. Mag.,’ 1896, vol. 42, p. 407.

I I call SY/SA more positive in one case than in another when it has either a larger positive value, or
a smaller negative value in the first case than in the second.

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

321

There are two ways in whieh the P.D, between the carbons, for a given current
and length of arc, may be lowered by the core ; (1) by an increase in the cross-
section of the vapour film, or the mist, or both ; (2) by a lowering of their specific
resistances. To see whether I could observe any change in the cross-sections, I have
traced a series of enlarged images of the arc with four sets of Apostle carbons,
using (1) + Solid — Solid, (2) + Solid — Cored, (3) + Cored — Solid, (4) + Cored
— Cored carbons.

The positive carbon was 1 1 millims. and the negative 9 millims. in diameter, and the
arc 2 millims. in length in each case, while the currents were 4, 6, 8, 10, 12, 14 amperes.
The diagrams were traced not only when the arc was normal in each case, but also
immediately after each change of current, so that the effect on the cross-section of the
arc of both an instantaneous and a normal change of current might be seen. Fig. 5 (p. 307 )
showed the first set of diagrams of the normal arc ; the others are too numerous to
publish, but the mean cross-sections of the purple part — the mist— in each, measured
as in fig. 5, may be found in Table III., those marked “non-normal” belonging to the
arc immediately after the change of current, and those marked “ normal ” to the arc
after all the conditions had become steady again.

Table III. — Mean Cross-Sections of Mist between fi- Solid — Solid, + Solid — Cored,
-f Cored — Solid, and -f Cored — Cored Apostle Carbons, 11 millims. and
9 millims.

Length of Arc, 2 millims.

Current in
amperes.

Normal.

Non-normal.

S.S.

S.C.

C.S.

C.C.

S.S.

S.C.

C.S.

C.C.

4

4-8

6-95

4-0

3-3

6

9-8

8-3

6-05

5 • 6

9-5

. 8-4

6-25

3-5

8

16-2

14-2

11-0

8-9

17-6

11-1

12-0

5-8

10

23-4

20-75

13-55

11-9

21-5

19-0

18-7

11-1

12

34-9

27-6

17-7

16-55

34-1

26-9

20-1

16-7

14

41-2

35-0

24-5

20-0

—

39-4

—

18-7

With a single exception, every number in each set is smaller than the corresponding
number in the preceding column. Hence, with both normal and non-normal arcs the
mean cross-section of the mist, for a given current, is largest when both carbons are
solid, smallest when both are cored, and is more diminished by coring the positive
than by coring the negative. Fig. II, besides showing well this marked difference in
the influence of the cores, makes it apparent that the difference increases, in every
case, with the current, for such currents as were there used,
yol. cxcix, — a, 2 t

322

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

We cannot measure the cross-section of the vapour film directly, hut, for a constant
length of arc, it must he roughly proportional to the cross-section of the mist where
it touches the crater. These cross-sections, which are given in Table IV., do not,
naturally, vary nearly so regularly as the mean cross-sections, hut still we can judge
pretty well what are the effects of the various cores. Coring the positive carbon,
for instance, distinctly diminishes the cross-section of the vapour film ; for every

Fig 11. Curve connecting the mean cross-section of the arc mist with the current for + solid, - solid,
-f solid - cored, + cored - solid, and + cored - cored carbons respectively. Apostle carbons, 11 millims.
and 9 millims. Length of arc, 2 millims.

number in column (7) is less than the corresponding one in column (5), and all hut
one in column (3) are less than those in column (1). Coring the negative carbon, on
the other hand, only seems to affect the cross-section that the vapour film
assumes immediately after a change of current, for while in the non-normal section
each number in (6) is less than in (5), and in (8) less than in (7), in the normal
section the numbers in (2) are sometimes less and sometimes greater than those in (1),
and those in (4) are nearly all greater than those in (3).

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

323

Table IV. — Cross-section of Mist where it touches Crater, with -f Solid — Solid,
+ Solid — Cored, + Cored — Solid, + Cored — Cored Carbons.

Apostle Carbons, 11 millims. and 9 millims.

Length of Arc, 2 millims.

Current in

Normal.

Non-normal.

Amperes.

(1)

(2)

(3)

D)

V)

(6)

(D

(8)

S.S.

S.C.

C.S.

c.c.

S.S.

S.C.

C.S.

C.C.

4

2-9

7-8

3-2

2-9

6

G • 8

9-0

5-3

5 • S

10-9

G • 25

G-25

3-6

8

16-0

13-0

10-9

14-4

1G-0

9-0

10-9

G-25

10

23-0

26-0

12-25

21-2

19-4

1G-8

15-2

15-2

12

32-5

25-0

17-6

22-1

31-4

23-0

17-6

21-2

14

39-1

36-0

23-0

24-0

—

33-6

—

19-4

Thus, taking Tables III. and IV. together, we find that a core in the positive
carbon keeps both the mist and the vapour film from being as large as they would be
with a solid positive, both immediately after a change of current and when the arc is
normal again. Coring the negative, on the other hand, while it has the same effect
on the cross-section of the mist as coring the positive, only diminishes the cross-
section of the vapour Jilin immediately after a change of current. If, therefore;
coring either carbon produced nothing but an alteration in the cross-section of the
arc, the resistance of the arc, and, consequently, the P.D. between the carbons would
be increased by the coring. It follows, therefore, that the diminution of the P.D.
between the carbons actually observed with cored carbons must be caused by a
lowering of the specific resistance of the vapour film or of the mist, or of both ; and
this lowering must be so great that it must more than compensate for the diminution
in their cross-sections.

It is easy to see howr the vapour and mist from a core in the positive carbon must
alter the specific resistance of the arc, but, since the negative carbon does not
volatilise, there seems to be no reason why coring it should have the same effect.
The core, however, consists of a mixture of carbon and metallic salts ; and metallic
salts have a lower temperature of Volatilisation than carbon, so that these salts may
easily be volatilised by the mist touching them, and, mingling with it, lower its
specific resistance.

Now take the fact that, with a constant length of arc, on increasing the current the
P.D. always diminishes less if the positive carbon is cored than if it is solid, and that
the reduction of diminution is sometimes so great that the P.D. remains constant for
a large increase of current, and sometimes even increases somewhat, instead of steadily
diminishing, as it does when both carbons are solid.

2 T 2

324

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

Every increase of current, whether the carbons are solid or cored, entails an
enlargement of the cross-section of the arc, and a consequent tendency of the P.D. to
diminish. While the current is so small, with cored carbons, that the volatilised
surface does not completely cover the core, the increase of cross-section is unaccom¬
panied by any change in the specific resistance of the arc. When the current is so
large, however, that the solid carbon round the core begins to volatilise, each increase

Current

Fig. 12. Curves exemplifying the changes introduced into the curves connecting P.D. with current for a

constant length of arc by coring the positive carbon.

of' current is accompanied by two tendencies in the P.D., the one to fall , on account
of the larger cross-section, the other to rise , because of the higher specific resistance
of the vapour and mist coming from the solid portion of the carbon. The curve
connecting the P.D. with the current must, therefore, be compounded of two. One,
such as A B C (fig. 12), which would connect the P.D. with the current if the positive

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

325

carbon were composed entirely of core, and the other, D E F, connecting the rise of
P.D., due to the increase of the specific resistance, with the current. The curve
connecting the true P.D. with the current is found by adding each ordinate of D E F
to the corresponding ordinate of A B C, as indicated in the dotted line. Whether
this resulting curve has the form G H Iv, or M N P, or Q B S (fig. 12) depends,
evidently, upon the relation between the increase of the cross-section and the rise of
specific resistance, i.e., on the relative structures and cross-sections of the core and
the outer carbon.

The fact, already obtained from Table IV., that, for the same current and length
of arc, the vapour film, and, consequently, the crater, is smaller with a cored than
with a solid positive carbon, explains why the arc can carry such a much larger
current without hissing when the positive carbon is cored than when it is solid. For
I have shown* that hissing is the result of that direct contact of the crater with the
air which follows when the crater grows too large to cover the end only of the
positive carbon and so extends along its sides, and this must happen with a smaller
current the larger the crater is with a given current, i.e., it must happen with a
smaller current when the positive carbon is solid than when it is cored.

How the Change in the Cross-Sections of the Mist and the Vapour Film, due to
a Change of Current, is Affected by Coring Either or Both Carbons.

Consider next the facts concerning the influence of the cores on the values
of SV/SA, when a small alternating current is superimposed on a direct current
normal arc, that the resistance of the arc is affected by this superposition. Here we
have to deal, not with the whole P.D. between the carbons, but with the change
in that P.D. that accompanies a given small change of current, and I shall show that
the effect of the core on this change must always be to add a positive increment
to SV/SA, the amount of which depends on the value of the direct current, the
length of the arc, and the frequency of the alternating current.

The influence of the core on the value SV/SA is two-fold ; it alters the amount by
which the cross-sections of the vapour film and the mist change, with a given change
of current ; and it makes their specific resistances vary with the current. We will
take each separately, the change of cross-section first. I shall call the part of SV/SA
that depends on the change of cross-section SVr/SA, and the part that depends on
the variation in the specific resistance 8V,./SA, so that 8VC/SA -f 8 VyS A = SV/SA,

I have already pointed out (p. 306) that if, when the current is increased, the ratios
of the new cross-sections of the mist and the vapour film to the old are greater than

* “The Hissing of the Electric Arc,” ‘Journal of the Proceedings of the Institution of Electrical
Engineers,’ 1899, vol. 28, p. 400.

326

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

the ratio of the new current to the old, then the resistance of the arc must have been
diminished more than the current was increased, and that hence the P.D. must have
diminished as the current increased, and SV/SA must be negative (provided always
that the specific resistance of the arc had not been altered). Similarly, when the
ratio of the cross-sections are smaller than that of the current SV/SA must be
positive.

In order to see the effect of the cores on these ratios, in the experiments of which
the results are given in Tables III. and IV., Tables V. and VI. have been drawn up,
in which the cross-section ratios are found by dividing the cross-section for each
current by the cross-section for the next smaller current ; and the current ratios by
dividing each current by the next smaller current. For the non-normal ratios
the larger cross-sections were taken from the non-normal sets in Tables III. and IV.
and the smaller from the normal , because it is the effect of the core when the
alternating current is superimposed on a normal arc that we are considering, and
because also the non-normal numbers in these Tables were found by suddenly
increasing the current when the arc was normal. For the normal ratios both currents
were taken from the normal sets.

For instance, the normal cross-section for a current of 8 amperes with the + solid-
— cored carbons in Table III. (p. 321) is 14 % and the non-normal cross-section for
10 amperes is 19'0, while the normal cross-section for the same current is 2075.
Thus, when the current is increased from 8 to 10 amperes, the current ratio is
-8- = P25, the non-normal cross-section ratio with these carbons is = P34, and
the normal is - °4 7p = P46. In this case, therefore, SVC/SA was negative, both when
the change of current was non-normal and when it was normal, for both the cross-
section ratios, P34 and 1'46, are greater than 1"25, the current ratio. Of course to
imitate the effect of an alternating current completely it would be necessary to
suddenly diminish the current as well as suddenly increasing it, but as this would
only alter the signs of both SV and SA, without materially changing their relative
values, it is not necessary for our purpose.

The non-normal ratios show the effect of the core on the change in the cross-
section, i.e., in the resistance of the arc, and therefore on 8V, when the frequenc}T of
the alternations is so great that the carbons do not change their shapes ; and the
normal, when it is so small that the changes are slow enough for the arc to rema n
practically normal throughout.

MRS, H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

327

Table V. — Ratios of Mean Cross-Sections of Mist taken from Table III.

Length of Arc, 2 millims.

Ratios of Mean

Cross-Sections,

Change

of

Current.

Current

Ratios.

Normal.

Non-normal.

S.S.

S.C.

C.S.

c.c.

S.S.

S.C.

C.S.

C.C.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

4 to 6

1-5

2-04

1-2

1-51

1-7

1-98

1-21

1-56

1-06

6 „ 8

1-33

1-65

1-71

1-82

1-6

1-8

1 • 33

1-98

1-04

8 ,, 10

1-25

1-44

1-46

1-23

1 ■ 33

1 • 33

1-34

1-70

1-25

10 „ 12

1-20

1-49

1-33

1-31

1-39

1 ■ 46

1-30

1-48

1-40

12 „ 14

1-17

1-18

1-25

1-38

1 -21

1-43

—

1-13

The most important point to observe in these tables is whether 8V0/SA is negative
or positive with each set of carbons, i.e., whether the cross-section ratios are greater
or less than the current ratios. Take first the non-normal ratios. When the positive
carbon alone is cored, the sign of 8VC/SA is decidedly negative, just as it is when both
carbons are solid, for all the cross-section ratios in column (9) of both Tables V.
and VI. are greater than the corresponding current ratios in column (2). Moreover,
with this particular length of arc, and these currents, the non-normal value of SV(./SA
does not appear to be altered by coring the 'positive carbon alone, for the non-normal
cross-section ratios in column (9) of each table are in some cases greater and in others
less than those in column (7). When the negative carbon alone is cored, the non¬
normal value of SYrjSA appears to be negative, but approaching the zero point ; for
in Table V., one cross-section ratio in column (8) is less than the corresponding
current ratio, one is equal, and three are greater; while in Table VI., three are less
and two are greater. When both carbons are cored, the non-normal value of 8Ve./SA
is positive; for three out of the five of the numbers in column (10) of Table V., and
the whole of those in the same column of Table VI. are greater than the corresponding
numbers in column (2).

328

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

Table VI. — Ratios of Cross-Sections of Mist where it Touches the Crater, taken

from Table IV.

Length of Arc, 2 millims.

Ratios of Cross-Sections at Crater.

Change

of

Current.

Current

Ratios.

Normal.

Non-normal.

S.S.

S.C.

C.S.

C.C.

S.S.

S.C.

C.S.

C.C.

(1)

(2)

(3)

(V

(V

(6)

(7)

(8)

(9)

(10)

4 to 6

1 -50

2-34

1-07

1-66

2-00

3-76

0-80

1-95

1-24

6 „ 8

1-33

2 • 35

1-44

2-05

2-50

2-35

1-00

2-06

1-08

8 „ 10

1-25

1-44

2-00

1-12

1-47

1-21

1-29

1-40

1 05

10 „ 12

1-20

1-41

0-96

1-44

1-04

1-37

0-88

1-44

1-00

12 „ 14

1-17

1-20

1-44

1-31

1-09

_

1-34

—

0-88

Turning next to the normal ratios, we find that when the positive carbon alone is
cored, SVff/SA lias still much the same negative value as when both carbons are
solid, since the numbers in column (5) differ on the whole very little from those in
column (3). When, on the other hand, it is the negative carbon alone that is cored,
there is a change, for instead of being a little below zero, SVySA is decidedly
negative, since in Table V. all but one of the numbers in column (4), and in Table VI.
all but two are greater than the corresponding numbers in column ( 2). When both
carbons are cored, there is an even greater difference between the normal and non¬
normal values of SVr/SA. For in Table V. all the numbers in column (6), and in
Table VI. all but two are greater than the corresponding current ratios, showing
that SV(./8A is negative for normal changes of current though it is positive for
non-normal changes with these carbons, currents, and length of arc.

To sum up the change in the value of 8 VySA produced by coring one or both of
the carbons, we find that while coring the positive carbon alone makes very little
difference in either the normal or non-normal change of cross-section that accom¬
panies a given change of current, coring the negative carbon diminishes the change
of cross-section, both for normal and non-normal changes of current, but more for
the second than for the first, and more when both carbons are cored than when the
negative alone is cored. Thus, coring the negative carbon both diminishes and
retards the change in the cross-sections of the arc that accompany a change of
current. This retardation of the change of cross-section is quite sufficient to account
for the fact already mentioned on p. 311, viz., that if 1 quickly altered the resistance
in the circuit outside the arc, when both carbons were cored , I could sometimes see

MRS. H. AYRTON ON THE MECHANISM OE THE ELECTRIC ARC.

329

the first quick swing of the voltmeter needle in the same direction as that of the
ammeter, but never when both were solid. For as the resistance did not alter
directly after the current, with the cored carbons, the new current would be flowing
through the old resistance for an appreciable time after the change, and so the
accompanying change of P.D. in the same direction as the change of current would
be able to influence the voltmeter needle.

The Change in the Specific Resistance of the Arc produced by a Change of Current

when Either or Both Carbons are Cored.

We have next to consider SV^/SA, the part of SV/SA that depends on the changes
in the specific resistances of the mist and the vapour that occur with each change
of current, when either or both carbons are cored.

As it is the positive carbon only that volatilises , while the negative simply burns,
coring the negative carbon alone must have a very different effect on the specific
resistance of the arc from coring the positive alone. For when the negative carbon
alone is cored, the whole of the vapour and almost the whole of the mist must issue
from the uncored carbon, the core in the negative carbon only contributing a little
metallic vapour to the mist in contact with it ; when, on the other hand, it is the
positive alone that is cored, the whole comes from the cored carbon. Thus, while,
with the cored negative, the vapour is always solid-carbon vapour, and the mist is
practically solid-carbon mist, with the cored positive the vapour and mist are both
core vapour and mist alone, until the current is large enough for the volatilising
surface to cover the whole core, and they only begin to have an admixture of solid-
carbon vapour and mist when the current is larger than this. When, therefore, the
negative carbon alone is cored, the specific resistance of the vapour is constant, and
that of the mist increases with each small increase of current, but more and more
slowly, with the same addition of current, the larger the original current before the
addition is made. The curve connecting SV^/SA with the normal current in this case
must, therefore, be of the form ABC (fig. 13), for the change of specific resistance
must be greatest when the current is just large enough for the mist to cover the
whole core, and must steadily diminish as the direct current increases after that, till
it becomes practically zero with very large currents, so that the curve becomes
asymptotic to the axis of current.

When the positive carbon alone is cored, the curve is quite different. If the arc
always remained perfectly central, it would be of the form D E F G (fig. 13). The
specific resistances of the vapour and mist would remain constant till the volatilising
surface was large enough to cover the core, so that until then SV,/SA would be zero,
and D E would be the first part of the curve. The first increment of current that
was added after this would increase the specific resistances by the largest possible

VOL. CXCIX. — A, 2 U

330

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC,

amount, because this would be the point at which the specific resistances of the
existing vapour and mist and of those added would be most different. Therefore the
curve would rise suddenly at E. After this, each addition to the normal current
would make the change of specific resistance due to the same added small non-normal
increment of current smaller and smaller, so that the curve would fall towards the
axis of current as shown in F G (fig. 13). Finally, there would already be so much
solid-carbon vapour and mist in the arc that the addition of a little more would
make practically no change, so that this curve also is asymptotic to the axis of
current. The fact that the arc is never really quite central, and that the volatilising
surface must therefore cover a little solid carbon long before it is larger than the
core, must introduce some modifications into the first part of the curve, shortening
D E, and making E F rise less abruptly, something like D E' F' G, but these
modifications are unimportant.

Fig. 13. Suggested curves connecting the part of SY/3A that depends on the change in the specific
resistance of the arc with the direct current for a constant length of arc.

When both carbons are cored, the curve must be like D E H K, or rather
D E" H' K, because the effect of the metallic vapour from the negative core will be
added to that of the positive core, and the change of specific resistance, when solid-
carbon mist begins to be added, will, therefore, be greater.

How the Whole Value 0/SV/8A is Affected by Coring Either or Both Carbons.

By combining the two changes in the resistance of the arc introduced by the core,
viz., that due to the changes in the cross-sections of the arc, and that produced by
the alterations in its specific resistance, we can see how the complete value of SV/SA
is affected by the core.

From what has been said on p. 329 it is clear that, if the cross-section ratios in
Tables V. and VI. can be considered typical, 8 Ve/8 A never has a greater negative
value when the positive carbon alone is cored than when both are solid ; never a
greater negative value when the negative alone than when the positive alone is

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

331

cored, and never a greater negative value when both are cored than when the
negative alone is cored, the conditions as to current length of arc, &c., being the same
in all cases. But SV,/SA is zero when both carbons are solid, is greatest when both are-
cored, and has always some positive value, however small, when either carbon alone'
is cored. Consequently, when the superimposed alternating current alters the1
resistance of the arc, if all other things are equal, the sum of these two, i.e., SV/SA, is-
more positive when either carbon is cored than when both are solid, and most positive1
when both are cored.

The general effect on SV/SA of coring either or both carbons is given in the
preceding paragraph, but with a given root mean square value of the alternating
current SV/SA depends not only on the nature of the carbons, but also on the
frequency of the alternating current, the magnitude of the direct current, and the
length of the arc. To complete our knowledge of the influence of cores on the value
of SV/SA therefore, we must examine the effect they produce on the curves
connecting each of these variables with SV/SA when the others are constant. Take
first the curves connecting SV/SA with the frequency of the alternating current.

The Change Produced in the Curve Connecting SV/SA with the Frequency of the
Alternating Current , by Coring Either or Both Carbons.

ABC (fig. 14), which is copied from fig. 10, is the curve, connecting SV/SA with the
current-frequency for solid carbons. Since for moderate frequencies SV/SA is always
most positive when both carbons are cored, and more positive when one is cored than
when both are solid, the curve when both carbons are cored must resemble D E F,
and the curves for one carbon cored and the other solid must lie between ABC, and
D E F, but we have no means of knowing which of the two will start the higher. It
follows, therefore, that the frequency with which SV/SA becomes positive, if it is not
already positive, for normal changes of current (i.e., for frequency 0), must be lower
when one carbon is cored than when both are solid and lowest when both are cored.
Thus, with the same direct current and length of arc, SV/SA may be positive for all
four sets of carbons, as at the points C, P, K, and F, or positive for some and negative
for others as at B N H and E, or negative for all. Moreover, since the true resistance
of the arc is greatest when both carbons are solid and least when both are cored, and
smaller when the positive alone than when the negative alone is cored, the horizontal
part of the curve, which shows the true resistance of the aj/c, must be highest when
both carbons are solid, next highest for + solid — cored carbons, lower for -j- cored
— solid carbons, and lowest of all when both are cored. Hence, since the curve for
two solid carbons starts lowest, it must cut all the others at some fairlv high

2 u 2

332

MRS. H. AYRTON OX THE MECHANISM OF THE ELECTRIC ARC.

frequencies, and that for two cored carbons, which starts highest, must also cut the
other two, so that the curves will be like I (fig. 14), if the curve starts lower when

Fig. 14. Suggested curves connecting SV/SA with the frequency of the superimposed alternating current

for a constant direct current and length of arc.

the positive alone is cored, than when the negative alone is cored ; and like II
(fig. 14) if it starts higher.

The Effect Produced by Coring Either or Both Carbons on the Curve Connecting the
Non-Normal Value of SV/SA with A, when the Length of the Arc is Constant.

Take next the curves connecting SA7 /BA with the normal direct current, when the
length of the arc and the frequency of the alternating current are constant.

In Tables V. and VI., pp. 327, 328, the cross-section ratios for solid carbons differ
less, on the whole, from the corresponding current ratios the larger the current on which
the increase of 2 amperes has been superimposed. This shows that with solid carbons,
when the length of the arc is constant, SA" SA diminishes as the current increases.
Consequently the curve connecting S\" SA with A for solid carbons is of the form
ABC (fig. 15). AVith cored carbons the curves depend not only on SAT/ A, which
is obtained from Tables A7, and A7I., but also on SAy/SA, the curves connecting which
with A are given in fig. 13. The curves connecting SVr/SA with A cannot be obtained
straight from Tables A", and VI., because the values are too irregular, but we can

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

333

deduce them from what we already know. For instance, when the positive carbon
alone is cored, it must have the same form A B C, as when both are solid, since the
change of cross-section due to a given change of current is not materially altered by
coring the positive carbon alone. Coring the negative carbon alone diminishes the
negative value of SVC/SA, and must diminish it most when the current is least, for it
is then that the metallic vapour from the core will be expended on the smallest

Current.

Fig. 15. Suggested curve connecting the part of SV/8A that depends on changes in the cross-section of

the arc with the direct current for a constant length of arc.

quantity of hard carbon mist, and will consequently have most effect. Hence the
curve for a cored negative and solid positive carbon must resemble D E F (fig. 15),
and the current for which SV^/SA becomes positive, if any, will depend upon the
length of the arc and the frequency for which the curve is drawn. Finally, with both
carbons cored, SVf/SA is even more positive than when the negative only is cored, so
that the curve must resemble G H K (fig. 15), since the same reasoning as before
shows that the cores have least effect when the current is largest.

To find the full curves connecting SV/SA with A, for each pair of carbons, vve have
only to add each ordinate of each curve in fig. 13 to the corresponding ordinate of the
curve for the same carbons in fig. 15. Curves resembling those that would be thus
obtained for one length of arc and frequency of alternating current are given in
fig. 16. The exact distance of each above or below the zero line, and the exact points
where it cuts that line must, of course, depend upon the length of arc and frequency

334

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

of alternating current for which the curves are drawn, but their relative shapes and
positions must he similar to those in fig. 16 whatever the length of the arc and the
frequency.

Fig. 16. Suggested curves connecting SV/SA with the direct current for a constant length of arc.

The Effect Produced by Coring Either or Both Carbons on the Curve connecting the
Non-Normal Value of SV/SA with the Length of the Arc when A is Constant.

Finally, we come to the curve connecting SV/SA with l, the length of the arc, when
the frequency of the alternating current and the value of the direct current are both
constant.

We must refer first of all to the connection between SV/SA and l when both
carbons are solid, in order to see how this connection is varied by the cores. P Q
(fig. 17) is the rise of P.D. that would accompany the increase of current SA, with
an arc of l millims., if the resistance of the arc did not alter with the current. Q It
is the fall of P.D. due to the enlargement of the vapour film and the mist (the
frequency of the superimposed alternating current is taken too great for the carbons
to be able to change their forms). When the current increases from A to A + SA,
therefore, the P.D. actually falls from P to 11. Now the rise P Q depends only on
the amount by which the current is increased, and the resistance through which that
increased current has to flow, i.e., on SA, A, and l ; or, since A and SA are supposed
to be the same for each length of arc, P Q depends simply on l, and increases when
l increases.

MRS. IL AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

335

The fall of P.D. — Q P — is more complex. It depends principally on how much of
the extra carbon volatilised by the larger current remains between the carbons, and
how much escapes along them. When the carbons have short thick ends more will
remain than when they have long pointed ones, and as the ends of the carbons are
thicker, with the same current, the longer the arc, a
small increase of current will diminish the resistance of
the arc more, the longer the arc. But the blunting of
the carbons, which is a rapid affair when the arc is short,
takes place more and more slowly as it is lengthened, till |
at last the addition of a millimetre or so to the length of ^
the arc makes practically no difference in the shapes of |
the carbons. Hence the diminution of resistance due ^
to the addition of SA to the current increases rapidly at q
first, when the arc is short, and more and more slowly
as the arc lengthens, till finally it becomes practically
constant ; and hence also, Q R — the fall of P.D. accom¬
panying this diminution — increases more and more slowly
as the arc is lengthened. Thus, while the rise of P.D. — P Q — increases at a constant
rate as the arc is lengthened, the fall, Q R, increases at a diminishing rate. While
the arc is so short, therefore, that Q II increases more rapidly than P Q when l is
increased, the whole fall of P. D. — P S — will increase, with the length of the arc, or,
since P S is — SV and SA is the same for all the lengths of arc, — SV/SA increases
as the arc is lengthened. When the arc is so long that P Q increases faster
than Q R, P S, and, therefore, — SV/SA will diminish as the arc is lengthened.
Between the two stages there must be a length ot arc for which — SV/SA is a
maximum. The curve connecting — SV/SA with l for a constant current, with solid
carbons, must, therefore, be of the form ABC (fig. 18), and there seems to he no
reason why , with very long arcs, SV/SA should not actually become positive, with
superimposed alternating currents of comparatively low frequency, even with solid
carbons.

The curves connecting SV/SA with /, when cored carbons are used, must resemble
the curve for solid carbons, ABC (fig. 18), but must be higher up the figure
(D E F, G H K) when one carbon is cored, and still higher up (M N P) when both
are cored. Also, since a change in the specific resistance of the arc must have more
effect on the value of SV/SA the longer the arc, the distance between the curves for
cored carbons and the curve for solid carbons must increase as the arc is lengthened,
as it is made to do in fig. 18.

Curves very similar to those in figs. 16 and 18 were obtained by Messrs. Frith and
Rodgers, in 1896, by actual measurements of SV/SA. Other measurements that
they carried out at the same time coincide with some of the other deductions I have
made concerning the influence of cores on the value of SV/SA. Hence, experience, as

336

MRS. H. AYRTON ON THE MECHANISM OF THE ELECTRIC ARC.

far as it goes, confirms the conclusions to which I have been led by a theoretical
consideration of the conditions. Since, therefore, all the principal phenomena of the
arc but one,# with cored and with solid carbons alike, may be readily accounted for

Length of Arc.

Fig. 18. Suggested curves connecting 8V/8A with the length of the arc for a constant direct current

without recourse to any such unusual attributes as a negative resistance, or even a
large back E.M.F., it seems superfluous to imagine their existence without stronger
proof of it than has yet been obtained.

* The one exception is the fall of potential of some 8 to 11 volts at the junction of the negative carbon
with the arc. This may be a true back E.M.F.

INDEX SLIP.

Wildebman, Meyer.— On
Influence of Light.

Chemical Dynamics and Statics under the
Phil. Trans., A, vol. 199, 1902, pp. 337-397.

Acetylene Light, Arrangements for, of High and Constant Candle-power,
and for Measurement of same.

Wildeeman, Meyer.

Phil. Trans., A, vol. 199, 1902, pp. 337-397.

Dynamics and Statics. Chemical, under the Influence of Light.

Wildebman, Meyer.

Phil. Trans., A, vol. 199, 1902, pp. 337-397.

Photo-chemical Reactions, Telocity of.

Wildebman, Meyer.

Phil. Trans., A, vol. 199, 1902, pp. 337-397.

Pure Gases (Chlorine and Carbon monoxide), Method for Preparation of.
Wildebman, Meyer.

Phil. Trans., A, vol. 199, 1902, pp. 337-397.

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| 337 J

VI. On Chemical Dynamics and Statics under the Influence of Light.
By Meyer Wilderman, Ph.D., B.Sc. ( Oxon .).
Communicated by Dr. Ludwig Mond, F.R.S.

Received January 30, — Read February 13, — Received in revised form, June 24, 1902.

Contents. Page

Introduction . 337

Part I.

General Arrangements for preparation of Pure Chlorine and Carbon Monoxide and for

filling Reaction Vessel . 341

Preparation of Pure Chlorine . ... 347

Preparation of Pure Carbon Monoxide . 349

Part II.

Arrangements for an Acetylene Light of 250 candle-power of constant Intensity and

Composition . . 351

Arrangements for Measurement and Adjustment of Intensity of Acetylene Light .... 356

Remaining parts of the Apparatus . 363

_ . ' . _ . Part III.

Experimental Results : —

The Law of Chemical Dynamics under the Influence of Light . 388

The Induction and Deduction Periods of Energy and the Chemical Induction and

• Deduction Periods in Light . 389

The Influence of Small Traces of Air and Water . 391

The Law of Chemical Statics under the Influence of Light . 393

Appendix : — Relation of Results obtained to Thermodynamics . 395

Introduction.

The nature of the forces which come into play when substances react one upon
another chemically, is a problem which has specially engaged scientific minds during
the last century. During the second half of that period chemical statics and
dynamics have developed into a veritable science. The general law governing the
velocity of chemical reaction and chemical equilibrium in homogeneous systems is
now known as the law of mass action, and was to a great extent foreseen by
Berthollet.# In heterogeneous systems the law concerning the velocity of physical

* C. L. Berthollet, ‘Essai de Statique Chimique,’ 1803; Wilhelmy, 1850; Harcourt and Esson,
1866 ; Guldberg and Waage, 1867 ; van’t Hoff, 1878.

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1)R. MEYER WILDERMAN ON CHEMICAL DYNAMICS

or molecular transformation also proves to be of a general and simple nature ; the
velocity being directly proportional to the surface of contact of the reacting parts of
the heterogeneous systems and to the remoteness of the system from the point of
equilibrium.* The velocity of chemical reaction and chemical equilibrium in
heterogeneous systems represent no phenomena sui generis, the laws concerning
them being only combinations of the above two laws.f The laws relating to
equilibrium found their rational explanation and foundation in the thermodynamic
researches of Horstmann, and more fully in those of W. Gibbs and yan’t Hoff,
whilst the laws applying to the velocity of reaction in homogeneous systems are the
result of van’t Hoff’s thermodynamic considerations.

In all the above researches the phenomena of the velocity of chemical reaction and
of chemical equilibrium are the outcome of those intrinsic properties of matter,
always existent in and inseparable from it, which we usually call chemical affinity or
chemical potential. It is known, however, that a system can be brought into a state
of reaction, and that new systems and new equilibria can be formed, when energy
from an external source, such as light or electricity, is introduced into it. The effect
of an electric current upon a chemical system, e.g., is determined by Faraday’s law
of electrolysis, whilst the thermodynamic connexion between chemical and electrical
(and gravitation) energy has been developed by W. Gibbs.

The object of this investigation was to ascertain, if possible, the laws governing
the velocity of chemical reaction and chemical equilibrium when this is caused by the
introduction of light energy into the system. Is the velocity directly proportional to
the amount of the light energy introduced or absorbed by the system in the unit of
time, independent of the reacting masses or concentrations, i.e., is the law here
analogous to that of Faraday for electrolysis, or is the velocity of reaction some
function of the reacting masses ? What are the laws governing chemical equilibrium
as affected by light ? It is evident that to furnish an answer to the above problems
careful experiments bearing directly on the fundamental issues in question and
a careful theoretical consideration of the results so obtained are absolutely needed.
This is the more imperative as from the hundreds of reactions known to be caused or
influenced by light| not half a dozen can be found suitable for quantitative measure¬
ments.

It soon appeared that the chemical reaction chosen for the study of the laws of
chemical kinetics must be very simple and as far as possible uncomplicated by
secondary phenomena. The chemical action observed must be caused by light alone,
and stop when light is removed. Bunsen and Boscoe’s reaction (H2 + Cl2 = 2HC1)
seems to fulfil these conditions, but inasmuch as no change of volume takes place, no

* See M. Wilderman, ‘Zeits. Physik. Chem.,’ 1899, 30, p. 341, and especially ‘Phil. Mag.,’ 1901,
July, p. 50.

1 See ‘ Zeits.,’ loc. cit., pp. 363-382.

I See Edek’s ‘ Handbuch der Photographic.’

AND STATICS UNDER THE INFLUENCE OF LIGHT.

339

answer could be afforded to the question whether the velocity of reaction is a function
of the varying masses or concentrations of the combining substances or not, unless
the hj^drochloric acid formed is as rapidly removed as it is formed during the reaction.
A secondary reaction would thus take place which would complicate the principal
reaction. For these and similar reasons* Davy’s reaction (CO + CL = COCL) was
chosen for study. The reaction occurs only in light, the gases can he used in a dry
state, and the volume of the mixed gases changes, two volumes of the original
mixture producing one volume of the compound.

Since highly dried gases combine very slowly, it was obvious that an artificial
source of light must be employed, which should be at the same time of great actinic
power and susceptible of being maintained constant in its intensity for long periods.
At the suggestion of Dr. Ludwig Mond the acetylene light was finally decided on,
as it is rich in actinic rays and its spectrum closely resembles that of sunlight.
Apparatus was devised for producing a flame constant in intensity and composition.
Arrangements were made for measuring the intensity of the light with an accuracy
to OT per cent., and suitable methods chosen for the preparation of pure chlorine and
carbon monoxide. A preliminary study extending over two and a half years was
made of the conditions under which the experiments must be conducted so as to give
not only concordant but accurate results, i. e. , results free from constant errors. The
general arrangement of an experiment was as follows : — -

Pure, dry chlorine and carbon monoxide were freshly prepared in the dark, and
there introduced into a reaction vessel, connected with a manometer to indicate the
variation of pressure during the reaction. The reaction vessel was placed behind a
quartz window, in a water bath kept at a constant temperature, and exposed to a
powerful acetylene light. The acetylene light was kept of constant intensity and
free from smoke by means of a special generator, balance governor, regulating tap,
purifier and burner. The intensity of the acetylene light was measured by means of
a Rubens’ thermopile and the deflections of a galvanometer, and the observed values
standardized by means of a Clark cell and manganin resistances. The variations of
the pressure in the reaction vessel were read on the scale of the manometer by
means of a cathetometer. After applying corrections for the variation of
temperature of the bath, atmospheric pressure, &c., the experimental results
representing the rate of formation of carbonyl chloride from chlorine and carbon
monoxide were subjected to a theoretical investigation.

* See Mellor, ‘Journal of the Chemical Society,’ 1901, p. 227.

2x2

340

DR. MEYER WILDERMAN ON CHEMICAL DYNAMICS

AND STATICS UNDER THE INFLUENCE OF LIGHT.

341

PART I.

General Arrangements for the Preparation of Pure CL and CO, and
for Filling the Reaction Vessel. (See Diagram 1.)

A Topler pump is connected with one arm (1) of the T-tube, and a double¬
cylinder Fleuss pump with the second (2), the third (3) carries the manometer (4),
and is connected with the rest of the apparatus by tube (5). A 3-way tap connects
either (l) or (2) with (6).

On the left of the diagram the apparatus for producing pure chlorine is shown.
Tube (7) containing cupric chloride was heated by one small flame; the chlorine
formed passes through tube (8) containing phosphorus pentoxide which is connected
with the reaction vessel whenever it is to be filled with chlorine. Before filling the
reaction vessel with chlorine, the cupric chloride and phosphorus pentoxide tubes
were exhausted and heated till a vacuum of ,01 millim. Avas obtained. Chlorine was
produced by heating the cupric chloride tube, the chlorine then removed, and a
further supply prepared and directly used for the experiment.

On the right of the diagram the apparatus for producing pure carbon monoxide is
shown. Vessel (10) contains sodium formate and concentrated sulphuric acid, vessel
(9) contains a concentrated solution of potassium hydroxide. When the mercury
tap (11) is closed, the carbon monoxide formed bubbles through the mercury in
vessel (12). When the tap is opened it passes through the U-tube (13) containing
pieces of caustic potash, then through the long tube (14) containing phosphorous
pentoxide to the receiver (15), which was chosen of a large size so that the gas
should be as little contaminated with air as possible. This vessel (15) was closed by
the mercury taps (16), (17) and (6), and was further protected from contamination
with air by the tube between the taps (16), (6) and (17) being filled with pure
carbon monoxide. Special precautions were taken to keep the carbon monoxide in
(15), (14) and (13) pure ; while the carbon monoxide from the tube between (16), (6)
and (17) was from time to time completely removed by pumping and heating, and
the tube again filled with pure carbon monoxide from (15). Before filling the glass
bulb (18) with carbon monoxide, every vessel containing the gas was first evacuated
to about ‘01 millim., fresh gas was again prepared and immediately used for the
experiment.

In the middle of the diagram the reaction vessel (II) is shown, into which carbon
monoxide and chlorine are brought together for exposure to light. This consists of
a glass cylinder with ground flanges on which two parallel quartz plates are fixed,
one in front and the other at the back. A capillary tube (19) connects the
manometer E with the reaction vessel, and with the tube (20) through which the
chlorine gas is passed from the cupric chloride tube into the quartz vessel. The

342

DR. MEYER WILDERMAN OX CHEMICAL DYNAMICS

latter is also connected with the hulb (18), (21) and (22) being mercury taps. The

bulb (18) is connected with a manometer (E') and is filled first with carbon
monoxide. The pressure of carbon monoxide in (18) should be greater than the
pressure of chlorine in the quartz vessel (11). By opening the tap (21) the carbon
monoxide is allowed to pass into the quartz vessel (II). No chlorine, as direct
careful tests showed, passes under such circumstances into the bulb (18).

I now pass to a more detailed description of the apparatus used.

The Reaction Vessel (fig. II.), was a glass cylinder of 3’65 centims. diameter,
3‘6 centims. long (capacity = 4R2G cub. centims.), with flanges 6 miflims. wide at
each end (as in fig. II.). The flanges were very carefully ground so that the two
perfectly smooth quartz plates, when placed on them, fitted perfectly, and an
excellent vacuum could be obtained. The quartz plates, with parallel surfaces,
45 sq. millims., and about 2 millims. thick, were cut out perpendicularly to the optical
axis, and were optically pure. The front plate turned the plane of polarisation to
the left, the back to the right.

Since chlorine acts on any cement which could be used for the purpose of keeping
a vacuum, the quartz plates were placed directly on the well-ground flanges of the
glass cylinder, and melted Crookes’ cement, consisting of beeswax and resin
(proportions, 5 parts beeswax to 8 parts resin) put on outside where the outer edges
of the quartz plates met the glass flanges, care being taken that the cement was
properly melted and free from air. The corners of the quartz plates projecting over
the flanges and those parts of the glass flanges which remain uncovered by the
quartz plates were then covered up with melted Crookes’ cement, the whole being
heated and made smooth and firm by a very small flame. In this condition the
vessel can preserve a very high vacuum for a very long time. The cement was then
covered with a varnish of pitch in benzine to protect the cement from the action of
water, and allowed to become thoroughly dry The outside of the cylinder was also
covered with pitch so that light might only enter through the quartz plates.

In the capillary tube (23), at the bottom of the glass cylinder, an iron-nickel couple
was fixed to measure the temperature of the gas in the inside of the cylinder during
the reaction. Two very fine iron-and-nickel wires (about 0'1 millim. diameter) were
made considerably thinner at their ends -(0 '05 to 0‘02 millim.) by repeated alternate
immersions in nitric acid and in water. About 1 centim. of the two very fine ends
were twisted together and a trace of melted soft solder put on the extreme ends. On
bringing it near a flame, the solder melted and ran down the twisted wires and was
quickly shaken off, forming only a very thin film of solder between the wires. The
soldered ends were then washed and all but 2 millims. of the connected ends were
removed. The two wires were then covered with shellac and a thin layer of pitch.
The very thin double wire (perhaps 0‘2 millim.) was then passed through the very
narrow capillary tube (23) projecting about I millim. into the cylinder, and fixed at
the bottom in the capillary tube by a trace first of pitch, then of Crookes’ cement,

AND STATICS UNDER THE INFLUENCE OF LIGHT.

343

and then again of pitch. This thermopile was very sensitive to variations of tem¬
perature, and does not interrupt any essential portion of the light on its passage
through the quartz vessel to the back plate. The second iron-nickel thermocouple
was wound outside the tube as near as possible to the first couple in the glass
cylinder, the two iron-and-nickel wires being drawn up round the cylinder, and then
wound up round the capillary tube. These wires were first covered with shellac and
then with pitch varnish. The thermopile assumed the temperature of the air or gas,
owing to its extreme thinness, with very great rapidity. Subsequently a thin, glass
bulb of similar volume was used instead of the quartz vessel.

The capillary tubes used were of a very narrow diameter. The capillary tube E
of the manometer was necessarily somewhat wider, so that the mercury might
move easily, and rapidly assume equal levels in both arms of the U-tube. A length
of 155 ‘7 centims. of the capillary of one of the manometers contained 9 '476 grams of
mercury = 0'701 eentim. The total volume of the capillary tubes in the manometer
over the mercury during the experiments (20 to 50 centims , &c.) is very small in
comparison with the volume of the cylinder of the quartz vessel, which is over
40 cub. centims., amounting to only a few tenths of 1 per cent. The manometer
contained a layer of concentrated sulphuric acid (about 10 centims. long), enough to
protect the mercury from the chlorine. The concentrated sulphuric acid freed the
capillary tube from any specks of dust, thus enabling the mercury to move in it much
more easily. The acid must be heated in vacuo after it is brought into the capillary
to remove air and sulphurous acid. Only after the sulphuric acid has been heated in
vacuo can a very high vacuum be obtained which will remain constant for any length
of time. In filling the capillary tubes with the sulphuric acid and mercury, the part
of the apparatus between (22) and (19'), which can be separated from the other parts,
is kept almost in a horizontal position, the open tube of the manometer being con¬
nected with a tube dipping into a small beaker of mercury covered by concentrated
sulphuric acid. Tap (22) being closed, tube (19') is held over the mercury in the
concentrated sulphuric acid till a few centims. of sulphuric acid rise in the tube, and
then dipped into the mercury, allowing it to follow the sulphuric acid. If the mercury
column is broken by sulphuric acid, the column is driven out from the capillary by
forcing air into the vessel at (19') until most of the acid is expelled from the capillary,
when the tube is refilled. The capillary U-tube of the manometer was calibrated in
the usual way. Since concentrated sulphuric acid adheres to the glass, the sulphuric
acid column is shorter when the mercury is low and longer when it is high, especially
if the mercury is moving rapidly up and down. During the experiment upon the
velocity of reaction the movement of the mercury is so slow that uniform results
are obtained ; but a reading must be made of the levels of both mercury and
sulphuric acid.

The density of the sulphuric acid used was easily found from the heights of the
sulphuric acid and of the mercury in both arms of the manometer when exposed to

344

DR. MEYER WILDERMAN OX CHEMICAL DYXAMICS

the pressure of the atmosphere. The U-tube of the manometer was fixed on a glass
scale silvered on the back to avoid parallax, the readings of the scale were easily
made by means of a cathetometer to 0'05 millim.* As the experiments proceeded, it
was found that there was no possibility of protecting, by means of mercury taps, the
gas mixture in the quartz vessel from contamination with air for more than a few
hours, even when to the mercury taps capillary tubes were added containing the same
reacting gases as the quartz vessel, with a second mercury tap at each end. The
tubes of the quartz vessel on both sides were sealed for this reason by means of a
hand blow-pipe as soon as the quartz vessel was filled with the gas mixture. Later
on the quartz vessel had to be abandoned altogether, chiefly for the reason that it
could not be heated before filling the reaction vessel with the gases ; more reliable
results were then obtained with a thin bulb of very pure glass than with the
quartz vessel.

Filling the Reaction Vessel with Carbon Monoxide and Chlorine.

Having read the position of the meniscus of the sulphuric acid and of the mercury
on both arms of the manometer, the apparatus was exhausted, every part of it being
heated to expel the air which persistently sticks to the glass walls of the apparatus.
Since, however, the quartz vessel (owing to the Crookes’ cement and pitch) could not
be heated, the complete removal of the air was effected as follows : — Having connected
the pumps through (l) and (2) with (6), the taps (6), (17), (22), (21) and (28) being
open and the taps (16) and (11) closed, all the vessels (7), (8), (E), (II), (18), (E'),
(S), (R), (19), (3), (2), (1) and (4) were exhausted first by the Fleuss and then Topler
pump, and all, except the quartz vessel (II), were heated. The concentrated sulphuric
acid in (S) and over the mercury in (E) was also heated until no more gas was given
off. In this way a high vacuum was obtained. The taps (6) and (28) were then
closed and carbon monoxide passed from its reservoir (15) into the bulbs (S) and (18)
and to the quartz vessel (II) until the pressure was about 10 centims. The carbon
monoxide was then removed by exhaustion, (16) and (28) being closed until a good
vacuum was indicated by the Topler pump, when the vessels were again filled with
carbon monoxide. This was repeated several times till the air was completely removed
from the walls of the quartz vessel. Better results, however, were obtained when a
thin bulb of pure glass was used and the bulb heated during the evacuation. But
even in this case, after the first evacuation the bulb was filled with carbon monoxide

* The mercury meniscus in the short arm always remains as clear as a mirror. From its position and
an ordinary calibration of the capillary tube of the manometer, the variation in the mercury column can
be determined with even greater accuracy than 0'05 millim. The readings of the mercury meniscus in the
long arm of the manometer, together with the upper ends of the sulphuric acid columns in the two arms,
enables the length of the sulphuric acid columns to be measured with an accuracy much exceeding
0'05 millim. mercury, considering that the specific gravity of the sulphuric acid is only about 1-S5.

AND STATICS UNDER THE INFLUENCE OF LIGHT.

345

and evacuated to a high vacuum of about O'Ol millim. Taps (16), (17) and (21) were
then closed, and chlorine prepared in (7) till the pressure in the reaction vessel
became equal to about 10 centims. Tap (6) was connected with the removable
pump (M), and (M) and (R) heated and evacuated, carbon monoxide sent into it
from (15), again evacuated, and then (17) and (21) opened and the chlorine removed.
The taps (17) and (16) were then turned off, (6) connected with the pumps,
(R) evacuated, all vessels (13), (14), (15), (R), (S), (E') and (18) evacuated, and
carbon monoxide from (9) and (10) removed as far as possible, partly by opening
tap (11) and then removing the same from (15), &c., partly by allowing the freshly-
prepared carbon monoxide to bubble for some time in (12). The bulbs (15), (S) and
(18) were then finally repeatedly filled, at one atmosphere pressure or more, with
fresh carbon monoxide from (10) (prepared in the dark, only one incandescent lamp
at a distance being used). The quantity and pressure of the carbon monoxide
introduced into the glass bulb (18) is known from the volume of the bulb and from
the indications of the manometer connected with it. The tap (17) was then turned
off ; the reaction vessel was next completely protected from light. The tap (28)
was now turned on, and 1 or 2 centims. of the tube containing cupric chloride was
then slowly and cautiously heated (in the dark), so as to evolve chlorine and to allow
it to pass to the reaction vessel at a slow rate. From the indications of the
manometer the amount of chlorine introduced into the quartz vessel was known, and
the production of chlorine was stopped by removing the burner from the tube as
soon as the desired quantity of chlorine, which is very small, had been introduced
into the quartz vessel. The capillary tube was immediately sealed up with a hand
blow-pipe at i/ — where it had been previously drawn out. After the heated parts
had cooled down, the position of the meniscus of the mercury and of the concentrated
sulphuric acid was read in both arms of the manometer by means of a faint light,
as well as the temperature of the room and the barometric pressure. The manometer
(E') was again read, the tap (22) closed, the tap (21) opened, and the carbon
monoxide, which is at a greater pressure in (18) than the chlorine in the reaction
vessel, was slowly allowed to pass through the drawn-out capillary tube (at i„) to
the quartz vessel, so as to keep the SOjHo column over the mercury the whole time
long enough, and when the mercury no longer moved in the manometer (E), the
tap (21) was closed and the capillary tube sealed at iu — where it was previously
drawn out. The vessel was again allowed to cool and the temperature and pressure
readings again noted.

From the variation in the height of the manometer the quantity of carbon
monoxide introduced into the quartz vessel was known. The volume of the bulb (18)
is immaterial, if it is of sufficiently large size to allow o,f the introduction of any
desired quantity of carbon monoxide into the reaction vessel. The bulb (18) was
then taken off, the end with the tap (21) immersed in a beaker containing a solution
of potassium iodide, and the tap opened. It was found that with the above

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346

DR. MEYER WILDERMAN ON CHEMICAL DYNAMICS

arrangements, i. e. , with a thin capillary tube drawn out at [if and with an over¬
pressure of carbon monoxide in (18), not a trace of iodine {i.e., of chlorine) can be
detected in the bulb (18), even with such a sensitive reagent as freshly-prepared
starch solution. By leaving the tap (21) intentionally open for half an hour, only
small traces of chlorine could be detected in the bulb (18). This shows that the
diffusion of the heavy chlorine gas to the top through a very thin capillary into the
vessel containing carbon monoxide, which is a vacuum for chlorine,* is extraordinarily
slow. For this reason the method of filling the vessels must not he reversed,
i.e., we must not fill the quartz vessels first with carbon monoxide and then with
chlorine from the bulb, nor can we uniformly mix the two gases in the two vessels
by opening the tap (21), even when the capillary tube is large, a method adopted in
many similar investigations, but which was found, at any rate in this case, to be
wrong.

The Removal of Chlorine from Tubes (7) and (8) and from the Quartz Vessel.

Removable Pump. (See fig. 3, Table I.)

Before passing to the description of the methods of preparation of pure chlorine
and carbon monoxide, a few words must be added as to the mode in which chlorine
can be completely removed from the vessels before a new experiment is proceeded
with. The removal of carbon monoxide is a simple matter — this being done by the
Topler pump, but chlorine cannot be removed by the Topler pump, because even
small quantities of chlorine instantly attack the mercury. All attempts to protect
the mercury pump by inserting tubes with precipitated copper or mercury for the
absorption of the passing chlorine completely failed. This, however, was effected in
the following manner : — The vessel (M) was connected through (30) with the vessels
containing chlorine and was heated and evacuated through the taps (29) and (31),
connected with (15) by means of a Topler and Fleuss pump, and while the tap (30)
was turned off, the taps (29) and (31) were then turned off and the tap (30) turned
on. A great part of the chlorine passes from the vessels containing it into the

* In this research the velocity of combination of chlorine and carbon monoxide, as a function of the
reacting masses, had to be studied. A horizontal gauge, as used by Bunsen and Roscoe, could not be
employed, because very great variations in the reacting concentrations or masses of the gases, amounting
to 70-80 per cent., had to be studied. Thus a mercury manometer had to be employed. To be able to
carry out this research, in view of the chlorine attacking the mercury, advantage was taken of the
extremely slow diffusion of chlorine, which in concentrated S04IL is still smaller than into a vessel of
carbon monoxide, which is a vacuum for chlorine. If the column of the concentrated SO.JR over the
mercury in the manometer is taken long enough (10 centims.), and care is taken that the filling of the
quartz vessel with chlorine or carbon monoxide is very slow, so that the concentrated SO4H2 should not
remain on the walls of the capillary tubes, but have time enough to run down, then we find that the dry
chlorine will not attack the mercury for days, and even weeks. Sometimes we find after a longer time
that the mercury meniscus becomes a little dull, without, however, losing its shape, and without the
mercury losing its mobility, and without interfering with accurate reading.

AND STATICS UNDER THE INFLUENCE OF LIGHT.

347

vessel (M). The tap (30) was again shut and the india-rubber tube removed from
the tajD (31). After opening tap (29) and carefully allowing air to pass into (M), the
vessel (M) was detached from the rest of the apparatus at (32), where the ground
tube (N) perfectly fitted into the neck of (M). The chlorine was then removed from
(M) by blowing through a tube introduced into it through its neck ; (M) was now
replaced on the tube (N) and evacuated. Chlorine was aspirated into (M) and blown
out until a pretty high vacuum was obtained in all the apparatus. Air was then
allowed to pass into the vessels which contained the chlorine and the whole again
evacuated directly by the Fleuss pump, and finally to a high vacuum by the Topler
pump, in the ordinary way. Chlorine can now be again prepared from the copper
chloride in (7).

Experience has shown that the only and perfectly reliable way of getting pure
gases free from any contamination with air is to remove, before each experiment, the
chlorine and carbon monoxide (though they are apparently perfectly pure) from all
the heated vessels as completely as possible, and to immediately prepare perfectly
fresh chlorine and carbon monoxide. As soon as they are prepared they are sealed
up in the quartz vessel (or the glass bulb used instead of it) as quickly as possible.

The Preparation of Pure Gases.

A. Preparation of Pure Chlorine. (See (7) and (8) in fig. 1, Table I.)

In order to get chlorine free from any admixture of air and water vapour, which
prove to be most fatal to the gas mixture, and also of any other gas, the ordinary
simple methods could not be employed. It was therefore prepared either from
platinous chloride or cupric chloride in a vacuum. Cupric chloride, suggested by
Dr. Ludwig Mond, has great advantages in comparison with platinous chloride ; it is
very much cheaper, water can be easily removed from it, and there is no danger that
oxygen or hydrogen from the air will be absorbed by the residue, as is the case with
platinum. Besides this, occluded gases cannot be completely removed from the
platinum unless by exhaustion at a temperature higher than that at which platinous
chloride gives off chlorine. Cupric chloride, so called “ purissima,” always contains
hydrochloric acid. This, it was found, cannot be completely removed. The cupric
chloride was, therefore, prepared from precipitated copper and chlorine, taking all
precautions to avoid conditions which might contaminate the product with hydro¬
chloric acid. Finely divided copper, precipitated from a solution of copper sulphate
by means of zinc, was placed in a long combustion tube drawn out at both ends.
The hydrogen current, first washed and dried, was passed over the copper (heated to
a dull red heat) for several hours to remove the film of oxide. The tube was next
sealed at one end and exhausted, and air was then allowed to pass into the tube,
which was again exhausted and heated. The hydrogen was thus completely removed.

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348

DR. MEYER WILDERMAN OX CHEMICAL DYNAMICS

The end of the tube was then opened and a current of chlorine, prepared from
manganese dioxide (freed from carbonates) and concentrated hydrochloric acid, passed
through two wash-bottles of water and two bottles of concentrated sulphuric acid,
was passed over the reduced copper.

Chlorine combines with copper in the cold, but as the reaction progresses the
copper and tube become heated from one end to the other. The heat developed is
usually so great that the cuprous chloride formed melts to a cake. When the contents
of the tube have become green, the tube is broken, the mixture of cuprous and cupric
chlorides is powdered and placed in a Jena tube drawn out at both ends. The tube
is now heated for its whole length to about 250-300°, and chlorine is allowed to pass
over it for a long time, the tube being shaken from time to time, when more chlorine
is absorbed. The whole mass is then allowed to cool in the current of dry chlorine.
From the increase in the weight of the tube and the weight of the copper taken, the
amount of the cupric chloride formed can be calculated, and, if necessary, the
operation of passing chlorine over the heated mixture of cupric and cuprous chlorides
is repeated. There is no necessity for the whole mass to be transformed into cupric
chloride. A current of air is then drawn through the tube to remove the chlorine,
for the reason stated above, and then one end of the tube is sealed up.

Tube (7) with cupric chloride and cuprous chloride thus prepared is now ready to
be used for the experiment ; it is placed on a combustion furnace, one end of it being
connected by means of a piece of india-rubber tube used for vacua to tube (8),
containing phosphorus pentoxide, and the india-rubber covered with Crookes’ cement.*

* It was found that no tube of soft glass could be used, the atmospheric pressure outside pressing the
glass in at the places where the tube was heated. Thus a tube of hard glass had to be used. This, as
known, cannot be joined with the soft glass of which the glass cylinder of the quartz vessel and the
capillary tubes are of necessity made. Since it was found that air (oxygen) and water vapour are just
the gases which are most fatal for the gas mixture, the heating of cupric chloride in a vacuum was
inevitable, and no other method could be employed instead. Luckily, the amount of chlorine gas required
for each experiment was exceedingly small, and only about gram. mol. of cupric chloride had to he
decomposed for each experiment, i.e., only 1 or 2 eentims. of the tube (7) had to be heated, and, since the
filling of the quartz vessel with chlorine and carbon monoxide had to be carried out (for reasons given
above) slowly, and cupric chloride decomposes at a comparatively low temperature, this centimetre or two
of the tube had to be heated slowly and cautiously with only a comparatively small flame. Tube (7) was
heated at the sealed end, which is more removed from tube (8), and the glass of (7) is made to meet the
glass of (8). Under these conditions not only is the glass of (7) and (8), where they meet, quite cool, but
the tube (7) is so already, being 30 or 40 eentims. removed from (8). The gas passed the draAvn-out cold
tubes (7) and (8), where they meet, vuth the india-rubber collar on the top, for only about 10 minutes.
Under these conditions no traces of the action of chlorine uoon the india-rubber collar can be found on

Jl

cutting the same, or by chemical analysis. Chemical analysis, hoAvever, can give us little information
about small impurities AArhen the quantities of gas are so small as those Avhich I had at my disposal.
There, however, still remains a superior analysis for impurities, Avhen no chemical or physical method can
be of any more use, viz., the possibility of getting regular curves and a velocity constant. This analysis
showed that either the chlorine is absolutely free from any impurities, or that they are so small and of
such a kind as not to interfere with the phenomena under consideration.

AND STATICS UNDER THE INFLUENCE OF LIGHT.

349

Before the chlorine was prepared for the experiment the tube containing cupric
chloride, (7), and the tube (8), with phosphorus pentoxide, were connected with the
tube (20) leading to the quartz vessel (II), and heated and exhausted till the pressure
was reduced to not more than 0’01 millim In this way the air was first expelled from
all vessels and from the cupric chloride. The tap connecting all these parts with the
mercury pump was then shut, and 1 or 2 centims. of the tube (7) gradually heated
nearly to red heat and chlorine slowly evolved. When the manometer indicated
that the pressure of the chlorine was about 10 centims., the heating of (7), and with
it the formation of chlorine, was stopped. After removing the chlorine by means of
the removable pump, fresh chlorine gas was admitted. This process was repeated to
expel the last trace of air, when chlorine was again introduced and sealed up in the
tube.

B. P reparation of Pure Carbon Monoxide. (See (9), (10), (13), (14) and (15)

of figure (1), Table I.)

Carbon monoxide gas was prepared from sodium formate (35 gr.), and a mixture
of concentrated sulphuric acid (200 gr.) and water (100 gr.), the proportions
given by Lord Rayleigh. Carbon monoxide is produced when the mixture is
heated, and its formation is stopped when the mixture is cooled down to the ordinary
temperature, so that the same solution can be repeatedly used for the production of
carbon monoxide.

The experiment was so arranged that neither the vessels nor the liquids used for
the reaction contained any air. The sodium formate was introduced into (10)
through the neck (25), the stopper of which was so well ground that when covered
with vaseline it could stand a vacuum for any length of time when the pressure of
the carbon monoxide in the vessel was one atmosphere. Vessels (9) and (10) had
the U-tube (13) containing pieces of caustic potash, the long and wide tube (14)
containing phosphorus pentoxide and the carbon monoxide reservoir (15), as well as
the tube (R) between (16), (6) and (17), the purpose of which was to protect the
carbon monoxide in (15) from contamination with air, were all heated and completely
exhausted. During this taps (11) and (16) were open, and taps (24), (26) and (17)
were turned oil.

The air had next to be removed from the channels of the taps (24) and (26). This
was affected in the way shown in figure (4). After all the vessels (9), (10),
(13), (14), (15), &c., were evacuated to a high degree, a capillary tube ( t' ) with the
tap (T) at one end and a piece of india-rubber tubing (R) at the other drawn-out
end, was pressed tightly into the tube of the funnel of (10), and another similar
one into the funnel of (9). (T) was turned off, (24) opened. After this the same

was done with tap (26). Since the volume of the channels in the taps (24) and (26)
and the space above them is perhaps only 0'3 cub. centim., while the volume of all

350

DR. MEYER WILDERMAN ON CHEMICAL DYNAMICS

the vessels is about 1500 cub. centims., the pressure in all the vessels is only about
Off 5 millim. The taps (24) and (26) were then closed and tap (T) opened to allow
the removal of the tube (t'). A little water was brought over the taps at (24)
and (26). Vessels (9), (10), (13), (14) and (15) were then again brought to a high
vacuum, and it is evident that the air still contained in the channels of the

taps (24) and (26) (the pressure in them being 0T5 millim.,
and the volume about 0'02 centim.) could later on introduce,
when opened to allow liquid to pass through them to (9) or
(10), a contamination with air into the vessels of only about

0 15 x 0 0w ~~~ Taps (16), (6), (17) and

1500

= 0'0000002 millim.

Fig. 4.

(11) were now closed and a concentrated solution of caustic
potash in water, first boiled in a vacuum and freed from air,
and kept in an evacuated flask, poured into the funnel of (24).
Some of the caustic potash solution was forced into (9)
through tap (24), but not so as to reach the tube leading
from (9) to (10), to prevent the vapour pressure in (9) from
pressing the solution into (10). Into the funnel of (24) a
mixture of two parts of concentrated sulphuric acid and of one
part of water boiled out in vacuo was quickly introduced. About
250 cub. centims. of the sulphuric acid solution was rapidly
passed into (10), and a burner placed under it ; more caustic
potash solution was again passed into (9) until tube (32) in (9)
was covered about 2 centims., when the vessel (10) was rapidly
heated by passing a flame round it. In this way the caustic potash solution may be
prevented from passing into (10) through tube connecting (9) and (10), and it
a trace of it does pass into (10) it is subsequently neutralised by the sulphuric acid
and does not affect the result in any way. Vessel (10) must be heated on all sides to
avoid bumping. Tap (11) is turned off during the formation of carbon monoxide to
prevent the distillation of water into vessels (13) and (14) containing solid caustic
potash and phosphorus pentoxide. The carbon monoxide formed in (10) passes
through a solution of caustic potash in (9), leaving there any traces of carbonic acid
or of sulphurous acid, and it presses the mercury in (33) down until it begins to
bubble through the mercury seal in (12), passing from there through tube (34) to the
open air. The carbon monoxide is then pumped up at the end (35) till the pressure
in (8) and (10) becomes about 150 millims. . the carbon monoxide which is still being
formed bringing the mercury in tube (33) down again till it begins to bubble through
the seal in (12). Repeating this several times, we expel the last traces of air from
the vessels and liquids. The vessels (13), (14), (15), (5) and (18) were then tilled
with carbon monoxide. Tap (11) is partially opened, so that the carbon monoxide
should not bubble too rapidly through the solution of caustic potash and only slowly

AND STATICS UNDER THE INFLUENCE OF LIGHT.

351

through the concentrated sulphuric acid in the tilted vessel (S). After a short time,
when the mercury had risen in tube (33) about 10 centims., tap (11) was turned off
again. This is done because the formation of carbon monoxide goes on more quickly
and regularly as the pressure of carbon monoxide increases. When the newly formed
carbon monoxide again begins to bubble through the mercury, the tap (11) is again
opened and shut as before ; this is repeated until the pressure of carbon monoxide
becomes everywhere a little over one atmosphere, i.e., the mercury column in (33)
does not rise again when the tap (11) is opened, and carbon monoxide continues to
bubble through the mercury (if a still higher pressure is desired the tube at (35) is
closed). The taps (11) and (16) are now closed (open at 35), allowing the carbon
monoxide gas to bubble through the mercury and escape to the air. After removing
the burner from (10) it is rapidly and uniformly cooled on all sides by means of a wet
cloth, until the formation and bubbling of carbon monoxide through the caustic
potash ceases. The tube (4) is also filled with carbon monoxide at the same time in
order to protect, subsequently, the carbon monoxide in (15) from contamination with
air. The glass tubes of the different parts of the apparatus are directly joined
together without the use of india-rubber tubes, and all the taps have mercury seals.

In the funnels of (9) and (10) sufficient caustic jiotash solution and of sulphuric
acid solution is always left, and taps (24), (26), and (25) should be so exceptionally
well ground that even after many weeks (when covered with vaseline) the taps remain
quite transparent. The object of these precautions was that pure and fresh gas might
be prepared quickly for each experiment. Carbon monoxide gas, which had been
kept in the vessels for even small lengths of time, was never used. When an experi¬
ment was made, vessels R, S, (18), E, (15), (14), (13), &c., containing pure carbon
monoxide, were always first evacuated, heated, and the freshly-prepared carbon
monoxide passed directly into the bulb (18), allowing it to bubble slowly through the
concentrated sulphuric acid in (S) as described. The tap (22) was then turned off,
(21) turned on, and then the capillary tube (19) at in was immediately sealed up with
a hand blow-pipe.

PART II.

Arrangements for an Acetylene Light of about 250 Candle-Power of

Constant Intensity.

Generator. ( Arrangement for Constant Pressure .)

Acetylene generators, although they have some advantages over the ordinary gas¬
holder, have also some disadvantages ; firstly, because every time a fresh container of
carbide is used, a fresh portion of air from the container is mixed with the gas, and
though the volume of the container is very small in comparison with the volume of

352

DR, MEYER WILDERMAN ON CHEMICAL DYNAMICS

the acetylene gas in the gas-holder, still the mixture is undoubtedly a source of error,
and special precautions have to be taken to remove the first portion of the acetylene
formed from the container, so as to keep the composition of the gas during the
experiments as constant as possible. Secondly, the pressure of the gas from such a

Sketch of Acetylene Burner and Purifier, Thermopile, Reaction Vessel, Bath, Water Manometer, and
Cathetometer. — A.B., acetylene burner; P, purifier; R.Th., Rubens thermopile ; B, bath; R.V.,
reaction vessel ; M, mercury manometer of reaction vessel ; Cat., cathetometer ; W.M., water
manometer.

generator varies very considerably in comparison with that supplied from large gas¬
holders. The generator used was the so-called “ Incanto,” by Messrs. Thorn and
Hoddle, with variations of 10 per cent, in the pressure of the gas delivered. This
variation in pressure was found to he due to the strain of the floating halls on the

AND STATICS UNDER THE INFLUENCE OF LIGHT.

353

chain of the generator, but a modification reduced the variations to about Ik per cent.
The axis x (fig. 6) on which the ball L turns, was filed up till the movement of the ball
in the axis was easy, and (3, which guides the balls, as
well as the stem of L at a, were then filed and clamped
to give the ball a perfectly free movement up and down,
but not much side play. Sheets of lead (s) were then
fixed round the stem as near to the balls as possible, and
the weight carefully adjusted so that a small additional weight of about 10-15 grammes
placed near the balls should draw them quite down and open the valve, and that the
removal of this weight should bring the balls up again, closing the valve. The
modified generator gives, without a balance governor, for the heights of the bell out
of the tank, between 16 '5 centims. and 5 centims. (these are the limits within which
the bell of the gas-holder chiefly varies in its height during the production and con¬
sumption of gas), a variation of only l-5 millim. for 100 millims., i.e., of 1^ per cent.
These variations of the pressure were further reduced by the balance governor.

It was found that with a burner of such a candle power as I had to use (about
12 x 20 = 240 candles), a pressure of about 4 inches is necessary (instead of the
usual 2). For this the upper chamber of the gas-holder had to be almost filled with
water, leaving only space sufficient for the expansion of the water (in the winter care
must be taken that the water does not freeze when the gas-holder is in the open air ;
more free space or a salt solution must be used). The gas generator was placed on the
roof of the laboratory ; from the generator a lead pipe brought the gas to the room ;
first to the balance governor, from the balance governor to the regulating tap, from the
regulating tap to the water manometer WM, and then to the purifier ; here it passed
over the purifying substance of the lower cylinder, then of the higher cylinder, and
thence to the burner.

The Regulating Tap to counteract the different sources of error was of the ordinary
type. One tap, «, must always be quite open, when acetylene is consumed during
the experiments. By turning another tap, the quantity of gas passing per unit of
time to the burner is regulated, and the adjustment is indicated by the pointer on a
fixed scale of 90 degrees. The purpose of this tap was to adjust the supply of gas
to the burner so that the intensity of the acetylene light should, under varying con¬
ditions, always be kept the same. A series of conditions affect the intensity of the
acetylene flame and necessitate the use of a regulating tap, such as the variation in
the diameter of the outflow tubes in the nipples of the burner or the variation of the
atmospheric pressure, which though in one and the same day hardly ever varying more
than I or 2 millims., changes during a longer period considerably. 'There is another
source of error of an irregular but temporary nature which also necessitates its use :
the admixture of air when a new container is used. Evervthing was done to remove
this source of error at its source. When a new container was introduced, the tap
supplying water to the same was opened, and the cross-bar of the container pressing

VOL. CXCIX.--A. 2 Z

Fig. 6.

354

DR. MEYER WTLDERMAN OX CHEMICAL DYNAMICS

a metal plate on the india-rubber ring somewhat loosened. Water entered the
container, the tap was turned off, and the acetylene with the admixed air allowed to
escape between the metal plate and the india-rubber ring. This was repeated two or
three times. In this way the air was expelled from tire carbide containers so that
when the cross-bar was tightened up and the tap opened again, no appreciable effect
of the new container upon the acetylene ought to have taken place. With such
precautions this source of error quickly disappears, owing to the gas-holder, leading
pipes, balance governor, purifier, and burner containing a great quantity of pure
acetylene — as can be seen from the observed intensity of light read from the deflection
of the galvanometer. Variations, however, still existed, and sometimes required to
be adjusted by the regulating tap. Another possible source of error was the variation
in the composition of the gas, arising from the carbide used not being always of the
same quality (it is not certain that such a source of error does exist). This was
counteracted by the adjustment of the regulating tap.

These adjustments were always guided by the indications of the deflection of the
galvanometer.

Water Manometer. (See fig. 5.)

Since it was necessary to work with a pressure of about 4 inches when the tap of
the burner was shut, and of about 2\ inches when it was open, OT millim. variation
in the height of the upper and lower side of the water manometer would indicate
0'2 millim. in 25 millim., i.e., 0‘8 per cent, variation in the pressure of the gas. Its
chief purpose is to indicate in a quick manner, whether all the apparatus connected
Avith the supply of the acetylene gas to the burner is in good working order.

The Acetylene Burner. (See fig. 5.)

All attempts to get a light of a great candle power, which would remain of a
constant intensity and composition have been, as far as I know, unsuccessful up to
the present. We have now standard lights of 1 or 1 0 candle power, but we have
none of 200 or 1000 candle power, since the light of the arc varies considerably
both in intensity and composition. The object here was to obtain a light of, say, 200
or 500 candle power, or of any other intensity desired, which would remain constant
in its intensity and composition for any length of time, which could at any time be
easily adjusted with great accuracy to the desired intensity, and which could be used
with ease in the ordinary work of a laboratory. With the burners on the market
which were tried, the gas is always passed either through a very thin slit giving a
flat flame, or through two thin pinholes — the two gas streams meeting in one point,
and giving again a thin flat flame (24-30 candle power). This is done in order to
get the surface of the flame in contact with the air as large as possible, to obtain
complete combustion and as white a flame as possible. It was found that after

AND STATICS UNDER THE INFLUENCE OF LIGHT.

24 hours’ use — often even after much less — the candle power of the burner, either
with the slits or pinholes, was no longer the same, and very often the slit or
pinhole was already so much carbonised that the flame began to smoke ; on the
contrary, pinholes with a larger diameter, giving 40-50 candle power the pair,
require a very much longer time before they become carbonised and begin to smoke,
though even their candle power also diminishes with time. Since the form and even
the thickness of the flat flame changes continuously, we can only get a constant
light by cutting out a piece from the middle of the flame for a certain time, screening
all the rest of it. Since we cannot get one flat white flame, by means of several gas
streams directed to the same point, of more than 60 or 70 candle power, the increase
of thickness or of the size of the flame beyond the 70 candle power being always
accompanied by the formation of smoke, we could thus, in the best circumstances,
not get a constant flame of more than 20 candle power, a candle power not very
different from standards already existing. On the other hand, it was impossible with
several flat flames to get one light of great intensity in a small space, since the flames
cannot be placed very near to one another (owing to the form of the burner), and a
flame of 200 or 250 candle power (12-15 nipples) necessarily occupies a very large
area. If we further consider that each of the flames is different from the others in
size and form, and that the flame of any one burner soon changes in form and size,
and that only a small part of each could be cut off securely by a screen so as to give
a constant light for some time, it is evident that the number of lights or nipples
which would be required for a 200-250 candle power light to remain constant would
be about 40 or 50. Assuming that even the greatest care be taken in placing the
burners and screens on as small an area as possible, still the burner would occupy too
much space, and no point or line could be calculated from the different lights which
could theoretically be assumed to be the point or line from which the total light
was coming.

The burner which was ultimately constructed free from these difficulties is that
shown in the drawing (fig. 5).

The wide tube A of the burner is divided into four narrower tubes as shown,
leading at intervals of 90° into the channel of a hollow ring cut in the brass ring.
In the hollow brass ring (AB) 12 small brass pieces containing capillary tubes were
fixed, and in the end of the brass pieces nipples were fixed. Each nipple has one
round pinhole of about 15 or 20 candle power; the holes are parallel to each other,
and each gives a flame in the form of a straight thick line of a few millimetres
diameter and of a few centimetres in height. Bound the brass ring a brass jacket
is fixed, forming a hollow ring connected with the tubes r and v' . Water runs
continuously through the hollow ring of the burner, which thus remains cool, in
spite of the fact that flames of about 200-250 candle power are concentrated in a
very small area. Using this arrangement of the burner, we find that the acetylene
flame becomes perfectly pure and clear, and remains so for any length of time, even

DR. MEYER WILDERMAN ON CHEMICAL DYNAMICS

356

when the nipples have pinholes of only 15 or 20 candle power each. The clear
circular flame so obtained consists of lines interrupted by narrow air spaces. Having
all twelve lines in a narrow circle of only 1 or 1-j inch diameter, at equal distances
from one another, the direction of all the flames perpendicular, and the flames of
equal length, though, perhaps, of not quite the same thickness, we can safely assume
that the light comes from a perpendicular line drawn through a point near the centre
of the narrow circle.

Since the error arising- from small variations in the thickness of the lines cannot
possibly in this case be greater than 1-2 millims. in the distance, this can, for the
distance we have to use (| to 2 metres), be completely neglected. The variations
in intensity in the line of the flame proved to be due chiefly to variations in its top ;
by means of the chimney y, all the top parts of the line-flames are cut off. Owing
to the draught the lines are straightened and the air supply increased, while the
products of combustion are removed and a clear white flame, remarkable for its
constancy and brilliancy, is obtained. The screen and chimney y can be moved
higher or lower by means of the screw d, so as to get not only a constant flame with
the photometer, but a flame of the desired intensity, the rest of the adjustment of
the intensity being produced by means of the regulating tap. Besides the upper
part of the flame, the lower part was also cut out by means of an adjustable short
cylinder, which was also water -jacketed and connected with the water jacket of the
burner.

Thus, with the simple arrangements described (generator, balance governor and
burner), a source of light of 200-250 candle power (or 500 candle power and more,
according to the number of nipples employed) can be obtained, which will remain
constant, within l or 2 per cent., for a considerable time without regulation or
adjustmentA By means of the regulating tap and measuring instruments, the
intensity of the light can be adjusted with an accuracy to 0T per cent., and even
much less.

The General Arrangements used for the Measurement and Adjustment
of tee Intensity of the Acetylene Light.

The Principle of the Method. Acetylene as a Universal Standard from
0-I or 1 to 500 or 1000 Candle Power.

The light of the acetylene burner, which is placed at a certain distance from the
thermopile, is allowed to fall on the exposed junctions of the Rubens thermopile
(with 40 iron-constantan junctions), which is connected with galvanometer. A
second measurement is made directly after by means of a Clark cell and standard

* It should be noted that with the increase of the candle power of the burner, a greater pressure of the
gas (in the gas-holder, &c.) is necessary for getting the best conditions for the liame.

AND STATICS UNDER THE INFLUENCE OF LIGHT.

357

manganin resistances (manganin 100,000 ohms, 6 ohms in the shunt were used, as
shown in the diagram ; the coil of the galvanometer, also of manganin, is 3 ohms).
These resistances give, under the conditions of experiment, a sufficient deflection
(about fl- 15 centims. to — 15 centims.), so that the value of the deflection caused
by the acetylene light is always measured in standard units, independent of the

6 oh ms.

ii imm n n if i n in 1 1 mm

Scabs.

sensitiveness of the galvanometer, &c. (the diagram also shows that the difference
between the temperature in the quartz vessel immersed in the bath, owing to the
reaction which takes place in it, and that in the bath is also measured with the same
galvanometer by means of iron-nickel thermocouples ; this, however, has nothing to
do with the photometer itself).

The principle of the measurement of the light intensity consists in its deter¬
mination objectively by means of the deflection of the galvanometer and standard
units (Clark, manganin resistances). We believe that if two sources of light
(say acetylene gas) at a fixed distance, say 1 metre from the plane containing a
given thermopile (with a given number of iron-constantan junctions), in the same
relative position to the thermopile (the lines of the flames being parallel to the
line of the exposed junctions and seen in the same position from the narrow tube in
the double copper cylinder of the thermopile), give, or are made to give, the same
deflection in standard units, then the intensity of the lights must be the same,
provided they are quite pure and free from smoke and that the burner is cool. We
assume that the heat effects of any source of light upon the exposed junctions of the
thermopile, i.e., the rise of their temperature, will be directly proportional to the
amount of light falling upon them, i.e., to the intensity of the light. This point is
of primary importance. A careful theoretical investigation of it was absolutely
necessary, and the investigation confirmed the conceptions. The electromotive force
of the thermopile being thus directly proportional to the difference of temperature
of the thermo-junctions exposed to the light, and of those which are left in the dark,

358

DR. MEYER AYILDERMAN ON CHEMICAL DYNAMICS

the intensity of the light of a given source of light must he directly proportional to
the corrected deflection of the galvanometer (or to the tangent of the angle), and
can be measured by it.

It should be remarked that this objective method of measuring the intensity
of light by means of a thermopile (or bolometer), and of checking it by means of
a Clark cell and known standard resistances, measures the total heat energy
produced by the given source of light in the thermopile, and thus differs from the
other methods of measuring the intensity of light, the ordinary photometers,
based upon the physiological effect of the light upon the eye. No doubt there is a
distinct difference between the two ; an acetylene light which we physiologically
perceive, say of 16 candle power, is very much cooler than a 16 candle-power coal-
gas light. The temperature to which the exposed junctions of the thermopile will be
raised by the rays of flames of the same candle power, but from different sources of
light, will therefore be different, the more so as the colour and composition of
different lights are also different. Properly speaking two lights from two different
sources (say acetylene and coal gas, or arc light) can neither lie compared physio¬
logically on account of their different compositions, nor in the objective way by
means of a thermopile or bolometer, as given above, and no light can be compared
except with a light of the same nature. For comparison of two lights of the same
kind there can be no doubt that the objective method by means of a thermopile or
bolometer is by far more accurate and reliable than measurement in the physiological
way. As, however, for many practical purposes, the intensity of different lights has
almost always to be expressed in candle units, different sources have to be compared
and measured in this physiological way as far as it is practicable. In this photo-
meti ic work, at any rate, all the standard units for comparison, the small as well as
the large ones, ought to be correct multiples of one another, i.e., ought to be
all from the same source of light and of the same composition. Acetylene, as far as
we know, is the only source of light which gives reliable standards from very small
units, such as O'l or 1 candle power (when only a part of the line is used) up to very
great units, such as 500-1000 candle power (when many lines on a space of about
2 inches in diameter should be used), and, except the burners, the same arrangements
can be easily manipulated for fixing and adjusting all the standards in an absolutely
correct and objective manner, as described here.

The author now passes to the description of the thermopile and of the other parts
of the arrangements, indicated by the above diagram, which were used for measuring
the intensity of the acetylene light.

The Thermopile (or Bolometer). See fig. 5.

A detailed description of the thermopile used is given by Rubens in the
‘ Zeitschrift fur Instrumentenkunde,’ 1898, pp. 65-69, but a few data with regard
to it must be given here.

AND STATICS UNDER THE INFLUENCE OF LIGHT.

359

The thermopile has 40 thermo-junctions # of iron and constantan, on an area
of 2 cent inis. , 20 of them in one line being exposed to the light, and 20 (10 in one
line on each side of the line of the junction exposed to the light) remaining in
the dark. The wires used are very thin (OT to 0 15 millim.), and the places of the
junctions (those exposed to the light as well as those remaining in the dark, are
hammered to little thin round plates (0-5 to 0'8 millim. diameter) so as to increase
their sensitiveness to variation of temperature, and are made dead black. Before
the line of junctions exposed to the light there is a small and then a larger cone of
polished nickel, placed opposite and near the line of junctions, which causes a greater
quantity of light to fall upon the junctions. The electromotive force of the thermo¬
pile is O’OOIOG volt per 1° C.

In connection with the Rubens’ thermopile a Crompton’s dead-beat galvanometer
was used, in order to avoid the numerous disturbances experienced by galvanometers
other than that of the D’Arsonval type, and its sensitiveness was adjusted to get a
sufficiently great deflection of the spot of light — not to get the greatest possible
deflection, but to arrange the measurements of intensity so that after they were
brought to a great accuracy, say of 0T per cent., the measurement should be made in
an easy, steady, and reliable manner, and independent of numerous obstructive
influences lying beyond the thermopile itself. This meant that special precautions
had to be taken with both the thermopile and the galvanometer.

Whatever form is given to the thermopile, the next precaution, especially when a
sensitive thermopile (or bolometer) is used, is to protect it from the influence of the
surrounding medium, so as to secure concordant results. The junctions which are
exposed to light are necessarily much more exposed to all sorts of air currents than
those protected and covered from the light, and are also more subjected to the
influence of the variation of the temperature of the room. The unavoidable
continuous use of the acetylene flame to be measured by the thermopile from time to
time, produces air currents, set up by local differences in the temperature of the room.
The sensitiveness of the whole arrangement was necessarily so great that if the hand
was put before the thermopile at the distance of 1 or 2 decimetres, the deflection of
the galvanometer was considerable. This difficulty was evaded in the following
manner : —

The thermopile was enclosed in a cylinder of thin copper (about 1 millim. thick ;
the two circular sides of the front and back being made of the same thin copper.
In the centre of the front side a circular opening of about 4\ centims. was cut and in
this a copper tube of the same diameter (for the quartz plate) was fixed. In the
centre of the circular copper plate behind, a thick but very narrow tube (of about
3 or 4 millims. internal diameter) was fixed for the purpose of seeing and directing
the thermopile upon the light of the burner. The thermopile was fixed on a piece of

* Through an accident our thermopile had only 38 junctions.

.360

DR, MEYER WILDERMAN ON CHEMICAL DYNAMICS

ebonite in the middle of the air space of the inner cylinder, so that the larger cone
should be fully exposed to the light (except a little of the four corners, the whole
cone was exposed) and the thermopile everywhere equally removed from the cylinder.
To protect the thermopile from moisture, which is especially fatal to its iron wires,
a cylinder with calcium chloride was put into the inner cylinder. By means of the
tubes in front and behind, the copper cylinder was fixed in another larger copper
cylinder, and the space between was filled with 6 or 7 litres of water. Except the
small space of the larger tube in the front side, the narrow tube behind and
two narrow tubes on the top, filled with Faraday wax, through which the insulated
leads from the thermopile were drawn, the inner cylinder was thus enveloped on all
sides by a thick layer of water. The outer copper cylinder was entirely covered by
very thick sheets of asbestos. In this way any variation of temperature in the inner
cylinder was counteracted by the conductivity of the copper sides, surrounded
by water, while the thick sheets of asbestos and the large quantity of water between
the cylinders made any rapid change of the temperature of the water due to
alteration in the temperature of the room impossible.

It was found that if a continuous water current from the main was passed between
the two cylinders (even if the current was passed simultaneously in different places
between the cylinders as well as between the front and back sides of the cylinder,
and the water from the main was used only after half an hour or an hour, when the
temperature of the water from the main ought to be constant), very large deflections
of the g alvanometer, as much as several centimetres, were observed, and these varied
continuously though the thermopile was closed to light and should have given no
deflection at all. Obviously the temperature of the water from the main is never
quite constant, continually varying by several thousandths or even hundredths of a
degree Besides this, layers of different temperature may exist in the bulk of the
surrounding water owing to the fact that the temperature of water from the main
is lower than that- of the room, and that it must take a long time before the

6 or 7 litres of water in the cylinders are replaced by fresh water from the main,
i.e., the bulk of the water is exposed to the influence of the warmer temperature of
the room for a longer time, and this warmer water is being continually mixed with
the cold water from the main. This difliculty was finally overcome by leaving the 6 or

7 litres in the cylinders to assume the temperature of the room, giving up the stirring
altogether and replacing the same by the water current from the main. Through
this the difference between the temperature of the water and that of the room was
made very small, i.e., the warming of the water between the cylinders by the
surrounding temperature of the room was made exceedingly slow. It should be
observed that it is not the constancy of the temperature of the air space of the
inner cylinder which we require, but that the variation of its temperature should be
the same everywhere, and so slow that it should be in comparison with the velocity
of cooling or warming of the thermopile by the surrounding air in the inner cylinder.

AX1> STATICS UNDER THE INFLUENCE OF LIGHT.

361

exceedingly small, while all the junctions should be at the same temperature in
the dark.

There still remained other sources of error. In the first instance, there was the
quartz plate fixed in front of the larger tube closing up and protecting the inner air
space from the air currents of the room. The quartz plate is exposed on the inside
to the temperature of the air in the inner cylinder, on the outside to that of the
room. There was additional reason why the temperature of the water between the
cylinders and of the air space in the inner cylinder had to be brought to the
temperature of the room ; because it is evidently necessary that the thermopile
should be exposed to the same temperature on all sides. In the second instance, the
quartz plate and the inner air space between the same, the cone and the copper ring
upon which the quartz plate is fixed, are being heated by the rays of light while
measurements of the intensity of light are carried out, the rest of the inner cylinder
remaining unexposed.

To counteract these sources of error the air space exposed to the light was made
very small in comparison with the total air space of the inner cylinder. The narrow
circular copper plate (cp, fig. 5), on which the quartz plate (</) was fixed, was also
water-jacketed by means of the india-rubber tube (y), so that the four ends of the
quartz plate, equal to about one-third of the total surface of the plate, were directly
cooled by the narrow circular plate. In front of the quartz plate a large wooden
screen was placed, and to its back (not to be seen in this drawing) a copper
cylinder was fixed of about 1 inch in thickness and of the same diameter as the
outside copper cylinder of the thermopile, filled with water of the same temperature
as that contained between the two copper cylinders of the thermopile.

In filling all the apparatus with water from the main, the space between the two
copper cylinders was first filled with water through E. From the top of the outer
cylinder the water passes through the india-rubber tube (y) behind the quartz plates
and through the india-rubber tube (f) to the lower part of the cylinder which forms
the water screen. From the top of the water screen the water passes through an
india-rubber tube and pewter pipe ( p) back to the tank, and then the run of water
is stopped. The india-rubber tubes f and y allow the screen to move up and down,
and when it is down the water screen quite covers the whole front surface of the
copper cylinder of the thermopile, being removed from it by only about half a centi¬
metre. Direct tests of these arrangements showed them to be successful ; after the
light has been used and the quartz plates again screened, the deflection returns to the
same zero quickly (the thermo-electromotive force in the dark never exceeding about
2 millims.), and remains so for any length of time, and the same deflection is obtained
every time the screen is opened again. It takes only a fraction of a minute for the
spot of light to attain its maximum deflection or return to zero.

3 A

VOL. CXCIX-. — A.

302

DR. MEYER WILDERMAN ON CHEMICAL DYNAMICS

The Galvanometer, dc. (See fig. 7.)

With an acetylene flame of about 250 candle power it was possible to use a
Crompton’s dead-beat galvanometer instead of the more sensitive one made by
N alder Brothers.

One of the conducting leads p from the Rubens thermopile leads to the interrupter
E (see diagram p. 357), the purpose of which is either to interrupt the current, or to
bring the lead (|i) from the thermopile into connexion with the lead ( p ') conducting
to the galvanometer (g), or to connect the lead (X) from the nickel-iron thermocouple
of the quartz vessel with the lead (p') conducting to the galvanometer. From E the
wire passes to the reversing key, which is enclosed in an asbestos box, and from here
to the galvanometer (g).

The spot of light from an electric incandescent lamp, after jiassing through a
lens, was reflected from the galvanometer mirror upon a transparent celluloid scale.

Great difficulty was experienced in steadying the galvanometer ; the suspension of
the galvanometer in a box on an india-rubber band, and the placing of the box on a
very heavy stone, which again was placed in its turn on thick pieces of india-rubber,
gave vibrations of 1 to 2 millims., owing to the fact that the dark room the author
had to use was near machinery at work. It was ultimately steadied in the following
manner : — A soft thick copper wire was drawn from one wall of the room to the
other, and the wire was first stretched and shortened until it stood the weight of the
box containing the galvanometer suspended on two very thick and flexible rubber
rings. Inside the box the galvanometer (with the wooden plate on which it was
standing on pieces of copper) was suspended from one rubber ring on the hook
of a screw, which was passed through a hollow wooden cylinder on the top of
the wooden box and held by a nut, placed on a metal plate on the top of the
wooden cylinder. By raising the screw outside and turning the nut the galvano¬
meter could be brought to any height desired, and by carefully turning the screw
it could be placed at any angle, so as to get the spot of light at any required
place on the scale. The screws of the galvanometer were resting upon metal pieces
on the wooden plate, and adjusted so as to get the required sensitiveness by
weakening the magnetic field, i.e., by bringing the circular coil partially out of
the central iron core. It was further necessary that the whole arrangement of
the suspension of the galvanometer should not be upset by the necessity of fixing
the leads to the galvanometer, since it was impossible to ensure that greater
vibrations of the leads outside the box would never take place. For this reason
the galvanometer was not connected directly with the heavy leads ; these were
fixed to the wooden box, then for each lead three or four pieces of very thin
galvanometer suspension wire (each about 2 inches in length) were soldered to
two pieces of thick copper wire ; one of them was fixed to the lead and the other
to the terminal of the galvanometer. Thus the galvanometer became for the first

AND STATICS UNDER THE INFLUENCE OF LIGHT.

363

time quite freely suspended in the box, the fine suspension wires on the one hand
allowing free movement to the galvanometer wherever and whenever it may be
required, and, on the other hand, exerting no directing influence whatever on its mass.

The small vibrations of the walls of the room were thus allowed to affect the
copper wire at the ends, and were weakened almost to zero before they wrere
transmitted to the middle part of the wire. These vibrations in the middle part
of the wire were further weakened by the two rubber rings upon which the box
with the galvanometer was suspended, and, lastly, the galvanometer itself was
made independent both of the vibrations of the box and of the leads. In this way
excellent results were obtained, the movements of the spot of light when the scale
was removed from the galvanometer 1 or 1’2 metres not exceeding OT millim.
Having now, from the acetylene flame, a deflection of the spot of light of about
10 to 15 centims., each reading was brought (and with it the possibility of adjustment
of the intensity of light to the right or left) to an accuracy of about O'l per cent.,
an accuracy far greater than that required for the research.

Determination of the Value of the Observed, Deflection of the Galvanometer or of the

Intensity of Light in Standard Units. (Fig. 7.)

With the arrangement described above, the deflection of the spot of light depends
upon conditions which may easily vary according to the circumstance and time.
Assuming that the distance of the source of light from the thermopile is fixed, that
the flame is in the correct position, that the distance of the scale from the mirror is
fixed, still, if the india-rubber rings should become a little stretched in time, or
any similar accident happen, the sensitiveness of the galvanometer would vary. It
is, therefore, necessary that measurements of the intensity of light should be made
independent of variations in the sensitiveness of the galvanometer. It is further
desirable to be able to express at once the intensity of light in standard units
independently of any given arrangements of the photometer, &c. For this reason,
directly after the measurement of the light by means of the thermopile was made,
a second measurement was made with a Clark cell and manganin resistances, as
given in the above diagram, thus determining the value of the deflection, as caused
by the light, in units given by the Clark and manganin resistances.

The Remaining Parts of the Apparatus, &c.

The Bath. (Fig. 5, p. 352.)

The reaction vessel with the mixture of chlorine and hydrogen was immersed in
a water-bath and there exposed to light. The bath contained quartz windows, and
the reaction vessel was placed behind one of them, the manometer M (E in fig. 1)
remaning outside the bath. The volume of the gas in the capillary tubes and in the

3 A

o

A

364

DR. MEYER WILDERMAN OX CHEMICAL DYXAMICS

capillary of the manometer was only a few tenths per cent, of the volume of the gas
in the reaction vessel immersed in the bath, so that if the differences between the
temperature of the room and that of the bath were very great a small correction
was necessary.

Two mercury thermometers were immersed in the bath, one divided into degrees
centigrade which could be read to 0°'l, and the other a 0°T thermometer of
Beckmann’s type, with divisions about 3 millims. apart from one another, allowing the
temperature of the bath to be read to 0C-01. The thermometers were kept very near
to the reaction vessel ; the temperature of the gaseous mixture in the reaction vessel
with the quartz windows was indicated by the iron-nickel thermocouple inside the
same. Later on the temperature was calculated, no thermocouple being employed, to
ensure the gaseous mixture not being contaminated with any traces of other substances
during the reaction. Near the capillary tube of the manometer was a thermometer,
divided into degrees centigrade to indicate the temperature of the room.

To keep the temperature of the bath constant at any desired temperature, the
copper bath was large, containing about 70 litres of water. It was covered with
very thick sheets of asbestos, placed in a wooden box with an air space between it
and the sides of the box, which was jffaced in another wooden box with another air
space between them, while the top of the bath was covered with a wooden lid. At
the bottom a small circle was cut out of the wood and asbestos for a rose burner
2 centims. distant from the exposed circle of the copper bottom. The temperature of
the bath itself regulated the supply of gas to the burner. The liquid was thoroughly
stirred at frequent intervals. Since with all these arrangements the temperature of
the bath could not be kept sufficiently constant, owing to the heat absorbed from the
powerful acetylene light, the temperature of the bath was adjusted to the desired
degree by melting ice, especially when the temperatures required were below 25° or
30°, and by thorough stirring. In this manner the variations of temperature of the
bath were kept within as narrow limits as possible during the whole time of the
experiment. Each time the readings of the manometer and of the temperature of the
hath and of the reaction vessel had to be made, the bath was effectually stirred,
readings being taken two minutes later. During this period of two minutes the
temperature of the bath near the quartz vessel does not rise under the action of the
light more than 0C,01. Since for our purpose it is important to know the difference of
the temperature of the gas mixture at two different times, so as to be able to apply the
necessary correction, the above manner of making the readings each time two minutes
after the bath was well stirred, eliminated the error in the determined differences of
temperature almost completely. An investigation of the velocity with which the
mercury of the manometer assumes its maximum when the glass bulb or quartz
vessel is immersed in the bath showed that after two minutes the temperature of the
gas in the bulb is much less than 0°’0J removed from the convergence temperature,
the amount of our reading error. This convergence temperature of the gaseous

AND STATICS UNDER THE INFLUENCE OF LIGHT.

365

mixture is higher than the temperature of the bath, but always remains constant for
the same intensity of the same source of light falling upon the gaseous mixture,
provided that the reaction goes on so slowly that the heating of the system by the
heat of reaction can be neglected. We are also able from the temperature of the bath
to calculate the temperature of the gas mixture. Having once determined the
necessary elements for such a calculation (in a manner given by the author on several
other occasions)* from the velocity of cooling of the gaseous mixture by the bath and
from the velocity of heating of the gaseous mixture by the given source of light (at
the beginning of the induction period), a thermocouple was not introduced into the
thin glass bulb used instead of the quartz vessel, as it wTas better to make sure that
during the reaction no vapour of any kind could enter into the gaseous mixture from
the cement with which the thermocouple has to be fixed in the capillary of the vessel,
or from the shellac and pitch with which the wires of the thermocouple have to be
covered in order that they may be protected against the action of chlorine. Indeed,
the best results, as far as experience goes, wrere obtained when none of these
precautions were neglected.

PART III.

Experimental Results. (Tables I.-V.)

In the following tables the experimental data are given : —

No. is the number of the observation made.

r is the time at which the observation was made.

r'-r " is the time between two successive operations.

77 is the reading of the manometer E of the quartz vessel at the time r, read
with the cathetometer (38 divisions of the cathetometer scale = 1 millim. of
the manometer scale).

tt'-tt" is the rise of the manometer E during the time t"-t.
is the intensity of the acetylene light, i. e. , the integral intensity of the light
of all wavedengths contained in the same, expressed in millimetre deflection
of the galvanometer read on the scale at the time r, including the thermo-
electromotive force of the Rubens thermopile in the dark ; Th.E.M.F. gives
the thermo-electromotive force of the Rubens thermopile in the dark, read
on the scale at the time r.

i'-th.e.m.f'. gives the intensity of light at the time t'. A correction for the
deviation of this value from the average intensity of the light during the
whole time of the reaction can be applied to the velocity constant K, given
in Tables (II., III., IV. and V.), putting K directly proportional to the

* See “On Real and Apparent Freezing-Points,” by M. Wilderman, ‘Phil. Mag.,’ December, 1897,
pp. 474, 475.

366

DR. MEYER WILDERMAN OX CHEMICAL DYNAMICS

intensity of light. I omit, however, this correction, because the variations
in the values of Iv, especially when taken, as in our case, at small intervals,
are too considerable for the application of very small corrections to be of any
essential use.

Sid. gives the sensitiveness of the galvanometer used for the measurement of
the intensity of light, measured with standard units (manganin resistances
and Clark), and expressed in millimetre deflection of the galvanometer on
the same scale. From this the value of (i) is also known in standard units,
is the actual temperature of the bath on 1° thermometer at the time r.

tr is the temperature tB read on the 0°T Beckmann thermometer at the same
time r.

tman is the temperature of the room near the manometer.

Since the volume of the gas in the capillary tubes changes during the
reaction, owing to the rise of the mercury in the same, its value is during
the reaction from 0‘2 per cent, to 0'5 per cent, of the volume of the gas in
the quartz vessel or the bulb.

A variation in the temperature of the room =1° produced a change in the
height of the manometer from rvnr. ^ A5 percent. ^ ^otal

pressure of the gas.

0-2 x 760

amounts to — - ~—r

273 x 100

When the pressure of the gas = 760 millims., this
= 0'005 millim. to 0'012 millim. per 1° variation in

the temperature of the room. So long as we investigate only small
intervals of the curve, i.e., when the temperature of the room could not
change by 1°, no correction need appear in the tables, and tman_ need not
enter into the equation. When, however, the curve is investigated at
greater time intervals, a correction for tman_ can be usefully applied.

t'r~t"r is the difference in the temperature of the bath (and gas mixture) at the
time t' and t", read to 0°'01 on Beckmann’s thermometer.
t'r-t"r corr. is the correction in millimetre pressure, which is to be added to the
observed tt'-tt " for the variation of the temperature of the bath. This
value, when the expansion of the glass bulb is simultaneously accounted
for, equals l'S millim. for each 1° variation of temperature of the bath. The
value of the correction was found from direct observations on the manometer,
by bringing the bulb successively to higher temperatures. This correction
is especially important when small parts of the curve are investigated, and
becomes of smaller importance the greater tt'-tt " is, since the variations
of t remain almost constant during the whole time of the experiment.
li is the barometric pressure at the time r, read with the vernier ; the temperature
of the mercury is given in brackets.

AND STATICS UNDER THE INFLUENCE OF LIGHT.

367

h" is the variation of the barometer on passing from t to r", when the
correction for the temperature of the mercury is also made. This correction
must be added to the observed tt’-tt" ; it is for the same day, when the
barometer changes very little, of little importance ; on the contrary, it
becomes important for the ordinary changes of the barometer.

7 j" cmr, is the true variation in the height of the manometer, if the atmospheric
pressure and the temperature of the bath (or of the gas mixture) should
have remained constant the whole time, i.e., after the corrections for the
variations of the temperature of the bath and for the atmospheric pressure
were made. Results were not reduced to normal atmospheric pressure, since
this would not serve any purpose (see Table I.).

368

DR. MEYER AVI L DERM AN ON CHEMICAL DYNAMICS

Table I. — CO and CL (in the glass bulb).

The glass bulb was filled with the gases in the dark. The gases freshly prepared
mercury and of the cone. S04H2 (sp.gr. 1*84)) on the manometer was 505 '4 millims.,
temperature of the room 240,5. On leaving the glass bulb in the dark from Friday,
temperature of the room to 223,3. The manometer rose from 510 to 513‘5, i.e., by
of barometer and temperature of the room. Thus during three days no combination
Barometer, 763'2 ( 2 1 0 ‘ 7 ) , temperature of room, 220,3. Manometer fell to 51‘17, i.e.,
variation of temperature of the room and of the atmospheric pressure. The vessel
over the bulb and the temperature of the bath brought the light of the manometer
the centre of glass bulb = 62 ‘5 centims.

No.

Time of
observation,

T — T

in

minutes.

Indication
of the

manometer at
the time r,

f n

7 1* — 7 T

in

millimetres.

The intensity
of light in
millimetre
deflection of
galvanometer,
i

( + th.e.m.f.).

i

-th. e.m.f.

Sensitiveness
of galvano¬
meter
measured
with
standard
units in
millimetre
deflection,
std.

(right & left).

1st curve.

h. m.

minutes.

millims.

millims.

millims.

millims.

1

11 50

512-8

—

—

266

15

0

2

12 5

512 -S

195*

195

—

45

-0-1x2

(th.e.m.f. = 0)

3

12 50

512-7

—

—

—

110

-0-3x2

4

2 40

512-4

—

—

—

SO

- 0-5 x 2

5

4 0

511-9

—

—

—

35

0

6

4 35

511-9

206

201-5

—

70

2-6 x 2

(th.e.m.f. = 4 • 5)

7

5 45

5 1 4 ' 5

—

—

266

Next day

—

—

—

1-3x2

—

—

—

2nd curve.

8

10 33

515-8-364-9

—

—

267

19

2-3

9

10 52

517-0-363-8

197

197

—

10

2-4

10

11 2

518-3-362-7

—

—

—

10-5

2-4

* At the beginning, during the induction period, when no reaction is visible, no special

AND STATICS UNDER THE INFLUENCE OF LIGHT.

369

Experiment started 5tli July, 1901.

in the dark. The partial pressure of CL (from the variation of the height of the
and of CO was 110 millims., when the barometer was 76 4‘6 millims. (2 1°'7) and the
12 a.m., till Monday, 11 A.M., the barometer changed to 766‘85 (21°'4), the
(513‘5-510‘0) 2 = 7 millims. instead of 7 '2 millims., which corresponds to the variation
whatever took place in the dark. From Monday it was left till Tuesday, 11 a.m.
by (51 ‘35-51 ’17) 2 = 3‘6 millims. instead of 3'65 millims., corresponding to the
was then placed in the dark of 20°‘8. The pressure of the water column in the bath
to 512 8. Distance of acetylene light from Rubens’ thermopile =105 centims. ; from

/ //

TT ~ T? COD’. ,

The read

Calculated

the variation

The

tempera-

tempera¬
ture of
bath on

Variation
of the

Calculated
correction for

Height of
barometer,

variation
of the
barometer

of the
manometer
when

ture of
bath on 1°
thermo¬
meter,
tD.

the TV°
Beck¬
mann
ther-

temperature
of the bath
and of the
gas mixture,

t'r-t"r.

tr-t r

in millimetres,
which is to
be added to

h

at the time

T

(and tempera¬
ture).

in milli¬
metres,
h’ - li"
(corrected

corrections for
temperature
of the bath
and for

No.

mometer,

7T — 77* .

for tern-

atmospheric

tr-

perature).

pressure
were made.

o

o

O

millims.

millims. °

millims.

millims.

1st curve.

20-8

3-47

0 • 05*

1-8 X 0-05 = 0-09

763-2 (21-4)

0-06

0-15

1

- -

'3-52

0-04

1-8x0-04 = 0-07

—

0-17

0-04

2

—

3-56

0-36

0-65

—

0-41

0-46?

3

—

3-92

0-43

0-77

—

0-29

0-06

4

—

4-35

0-15

0-27

—

0-13

0-40

5

—

4-50

0-38

0-68

—

0-28

6-16

6

—

4-88

761-95(20-8)

7

—

—

-0-48

-0-86

—

- 1-8

-0-06

Next day

2nd curve.

21-7

4-40

0-03

1-8x0-03 = 00-5

763-75(20-8)

0-06

2-41

8

—

4-43

0-05

0-09

—

0-03

2-52

9

4-48

0-05

0-09

0-03

2-52

10

care was taken to keep the temperature of the bath and the intensity of light constant.
VOL. CXCIX. - A. 3 B

370

DR. MEYER WILDERMAN ON CHEMICAL DYNAMICS

Table I.- — CO and CL (in the glass bulb).

No.

Time of
observation,

T.

// /

T — T

in

minutes.

Indication
of the

manometer at
the time r,

/ n

7 T — 7 T

in

millimetres.

The intensity
of light in
millimetre
deflection of
galvanometer,
i

( + th.e.m.f.).

i

- th.e.m.f.

Sensitiveness
of galvano¬
meter
measured
-frith
standard
units in
millimetre
deflection,
std.

(right & left).

2nd curve.

h. m.

minutes.

millims.

millims.

millims.

millims.

11

11 12 i

519 • 5-361 • 5

204

202

9 -5

2'2

(th.e.m.f. = 2)

12

11 22

520-6-360-4

—

—

—

10-5

2-3

13

11 32|

521-8-359-3

203

201

—

10

2-0

(th.e.m.f. = 2)

14

11 42

522-8-358-3

204

202

—

10

1-9

(th.e.m.f. = 2)

15

11 52

523-8-357-4

203-5

201-5

—

10

1-9

(th.e.m.f. = 2)

16

12 2

524-7-356-4

—

—

_

10

1-7

17

12 12

525-6-355-6

—

—

—

10

2-3

18

12 22

526-8-354-5

202

200

—

10

1-5

(th.e.m.f. = 2)

19

12 32

527-5-353-7

—

—

—

10

1-6

20

12 42

528-3-352-9

—

—

—

9

1-7

21

12 51

529-2-352-1

—

—

—

9-5

1-5

22

1 21

530-0-351-4

204

201-5

—

9-5

1-4

(th.e.m.f. = 2)

23

1 12

530-7-350-7

—

—

—

10

1-4

24

1 22

531-4-350-0

—

—

- -

11-5

2-1

25

1 331

532-5-349-0

—

—

—

8-5

1-1

26

1 42

533-0-348-4

202

198-5

10

1-2

(th.e.m.f. = 3- 5)

27

1 52

533-6-347-8

—

—

267

10

1-0

28

2 2

534-1-347-3

—

—

—

10

1-4

29

2 12

534-8-346-6

202

198-5

—

10-5

1-3

(th.e.m.f. = 3 • 5)

30

9 22i

535-5-346-0

202-2

198-5

—

11

1-0

(th.e.m.f. = 3-5)

31

2 334

536-0-345-5

—

—

—

9

1-2

32

2 421

536-6-344-9

—

—

—

9-5

1-0

AND STATICS UNDER THE INFLUENCE OF LIGHT.

371

Experiment started 5th July, 1901 — (continued).

The

tempera¬
ture of
bath on 1°
thermo¬
meter,

The read
tempera¬
ture of
bath on
the Ty
Beck¬
mann

ther¬

mometer,

tr.

Variation
of the

temperature
of the bath
and of the
gas mixture,

t’r-t"r.

Calculated
correction for
t'r - t"r

in millimetres,
which is to
be added to

/ r/

7 T — 77“ .

Height of
barometer,
h

at the time

(and tempera¬
ture).

Calculated
variation
of the
barometer
in milli¬
metres,
h' - h"
(corrected
for tem¬
perature).

77 — 77 corr.,

the variation
of the
manometer
when

corrections for
temperature
of the bath
and for
atmospheric
pressure
were made.

No.

o

0

0

millims.

millions. o

millims.

millims.

2nd curve.

—

4-53

—

11

-0-06

-0*11

0-03

2-12

—

4-47

—

12

0-05

0-09

0-03

2 • 42

—

4-52

—

13

0-04

0-07

0-03

2-10

—

4-56

—

14

-0-03

-0-05

0-03

1-88

—

4-53

—

15

0-06

o-n

0-03

2-04

—

4-59

—

16

-o-oi

-0-02

0-03

1-71

—

4-58

—

17

-0-26

-0-47

0-03

1-86

—

4-32

—

18

0-04

0-07

0-03

1-60

—

4-36

—

19

/

0-07

0-13

0-03

1-76

—

4-43

—

20

-0-15

-0-27

0-03

1-46

—

4-28

—

21

0-09

0-16

0-03

1-69

—

4-37

—

22

0-05

0-09

0-03

1-52

—

4-42

763-3 (21 -8)

23

0-06

0-11

o-oi

1-52

—

4-48

—

24

- 0-22

-0-40

o-oi

1-71

—

4-26

—

25

0-08

0-14

o-oi

1-26

—

4-34

—

26

0-06

0-11

o-oi

1-32

2U7

4-40

—

27

0-08

0-14

o-oi

1-15

_

4-48

—

28

-0-15

-0-27

o-oi

1-14

—

4-33

—

29

0-05

0-09

o-oi

1-40

—

4-38

—

30

0-09

l-8x0-09 = 0T6

0-01

1-17

—

4-47

—

31

-0-14

-0-25

o-oi

0-96

—

4-33

—

32

0-05

0-09

o-oi

1-10

1

3 b 2

372

DR MEYER WILDERMAN OX CHEMICAL DYNAMICS

Table I. — CO and Cl2 (in the glass bulb).

Sensitiveness

of galvano-

The intensity

meter

Indication

of light in

measured

Time of

// •

T — T

of the

7 T — 7 t"

millimetre

with

No.

observation,

in

manometer at

in

deflection of

l

— th. e.m.f.

standard

T.

minutes.

the time t,

millimetres.

galvanometer,

units in

7T.

i

millimetre

( + th.e.m.f.).

deflection

std.

(right & left).

2nd curveJ

h.

m.

minutes.

millims.

millims.

millims.

millims.

33

2

52

537 • 1-344-4

—

—

11

i-i

34

3

rv

0

537-7-343-9

—

—

—

10-5

1-2

35

3

13-i-

538-3-343-3

202

198-5

—

11

1-0

(th.e.m.f. = 3- 5)

36

3

244

538-8-342-8

—

—

—

12

1-3

37

3

364

539-5-342-2

202

198-5

—

13-5

1-1

(th.e.m.f. = 3 • 5)

38

3

50

540-0-341-6

201

197-5

—

16-5

1-6

(th.e.m.f. = 3 -5)

39

I

540-8-340-8

202

198-5

—

13*5

1-2

(th.e.m.f. = 3-5)

40

4

20

541-4-340-2

204

199

—

15-5

1-1

(th.e.m.f. =5)

41

4

354

542-0-339-7

—

—

—

17-0

1-4

42

4

524

542-7-339-0

202

198-5

—

15

1-0

(th.e.m.f. = 3 • 5)

43

5

74

543-3-338-6

203

199-5

267

13

1-4

(th.e.m.f. = 3 - 5)

44

5

204

544-0-337-9

—

—

—

3rd curve.

11

0-5

45

5

314

544-2-337-6

—

—

—

15

0

46

5

464

544 • 2-337 ■ 6

—

—

—

10

0

47

5

564

544-2-337-6

—

—

267

Next day

_

• —

1-8

—

—

4th curve

48

10

56

545-0-366-6

202

200

265

10

0

(th.e.m.f. = 2)

49

11

6

545-0-336-6

—

—

—

10

0-4

50

11

16

545-2-336-4

201

199-5

—

10

0-6

(th.e.m.f. = 1 - 5)

AND STATICS UNDER THE INFLUENCE OF LIGHT.

373

Experiment started 5th July, 1901 (continued).

The

tempera¬
ture of
hath on 1°
thermo¬
meter,
t\ j-

The read
tempera¬
ture of
hath on
the Ty°
Beck¬
mann
ther¬
mometer,
t,.

Variation
of the

temperature
of the bath
and of the
gas mixture,
/' - f"

Calculated
correction for

t'r - t"r

in millimetres,

which is to

be added to
/ //

7T — 7T .

Height of
barometer,
h

at the time

T

(and tempera¬
ture).

Calculated
variation
of the
barometer
in milli¬
metres,
fi - h
(corrected
for tem¬
perature).

_ / /'

n ~~ 7T corr.,

the variation
of the
manometer
when

corrections for
temperature
of the bath
and for
atmospheric
pressure
were made.

No.

-

o

o

millims.

millims. o

millims.

millims.

2nd curve.

—

4-38

—

33

0-09

0-16

o-oi

1-27

4-47

—

' 34

-0-10

-0-18

o-oi

1-03

—

4-37

—

35

0-09

0-16

o-oi

1 - 17

4-46

—

36

-0-11

-0-20

0-02

1-21

—

4-35

—

37

0-11

0-20

0-02

1-32

—

4-46

38

-0-09

-0-16

0-02

1-46

—

4-37

—

39

-0-03

-0-05

0-02

1-17

—

4-34

—

40

0-12

0-22

0-02

1-34

—

4-46

—

41

-0-05

-0-09

0-02

1-33

—

'4-41

—

42

0-12

0-22

0-02

1-24

—

4-53

—

43

-0-16

-0-29

0-02

1-13

—

4-37

_

44

0-02

0-04

o-oi

0-55

3rd curve.

—

4-39

—

45

0-07

0-13

0-02

0-15

—

4-46

46

0-07

0-13

o-oi

0-14

—

4-53

762 -85 (21 -4)

47

—

—

-0-21

- 0 • 38

—

- 1-7

-0-28

Next day

4th curve.

—

4-32

764-55 (21-4)

48

o-oi

0-02

0-06

0-08

—

4-33

—

.49

0-03

0-05

0-06

0-51

—

4-36

—

50

0-06

0-11

0-06

0-77

374

DR. MEYER WILDERMAN ON CHEMICAL DYNAMICS

Table I. — CO and Cl2 (in the glass bulb).

No.

Time of
observation,

T.

n f

T — T

in

minutes.

Indication
of the

manometer at
the time r,

7 T.

t n

7 r — a

in

millimetres.

The intensity
of light in
millimetre
deflection of
galvanometer,
i

( + th.e.m f.).

i

- th.e.m.f.

Sensitiveness
of galvano¬
meter
measured
with
standard
units in
millimetre
deflection
std.

(right & left).

4th curve.

h.

ID.

minutes.

millims.

millims.

millims.

millims.

51

11

26

545- 5-336-1

—

—

_

13

1-0

52

11

39

546-0-335-6

203

200

—

21

2-3

(th.e.m. f. = 3)

53

12

0

547-2-334-5

—

—

—

18-5

1-3

54

12

184

547 • 9-333 • 9

202

200

—

16

1-2

(th.e.m.f. = 2)

55

12

34-|-

548-5-333-3

203-5

201-5

—

15-5

1-0

(th.e.m.f. = 2)

56

12

50

549-0-332-8

201-5

199-5

—

15

1-0

(th.e.m.f. = 2)

57

1

5

549-5-332-3

,

—

—

—

15

1-0

58

1

20

550-0-331-8

203-5

200-5

—

20

1-2

(th.e.m.f. = 31

59

1

40

550-6-331-2

202-0

198-5

—

20

1-4

(th.e.m.f. = 3- 5)

60

2

0

551-3-330-5

203-5

200-5

—

21-5

1-1

(th.e.m.f. = 3)

61

2

214

551-9-330-0

204

201

—

21-0

1-2

(th.e.m.f. = 3)

62

2

424

552-5-329-4

—

—

—

23-5

0-9

63

3

6

553-0-329-0

202

199

—

(th.e.m.f. = 3)

44

—

2-1

—

—

_

5th curve.

64

3

55

554-0-327-9

199-0

199

—

22

0-8

65

4

17

554-4-327-5

202

200

_

23

1-0

(th.e.m.f. = 2)

66

4

40

554-9-327-0

—

—

20

0-6

67

5

0

552-2-326-7

203

200-5

—

23

1-2

(th e.m.f. = 2 ■ 5)

*

68

5

23

555-8-326-1

202-5

200

—

22

0-8

(th e.m.f. = 2-5)

69

5

45

556-2-325-7

202-5

200

265

(th.e.m.f. = 2-5)

AND STATICS UNDER THE INFLUENCE OF LIGHT.

375

Experiment started 5th July, 1901 (continued).

/ n

tr — 7T coir..

The read

Calculated

the variation

The

tempera-

tempera¬
ture of
bath on

Variation
of the

Calculated
correction for

Height of
barometer,

variation
of the
barometer

of the
manometer
when

ture of
bath on 1°

the 7yJ

temperature
of the bath

t'r - t"r

in millimetres,

h

at the time

in milli-

corrections for

No.

thermo¬

meter,

G-

Beck¬

mann

ther¬

mometer,

and of the
gas mixture,

t y — t y.

which is to
be added to

t //

7T — 7T .

T

(and tempera¬
ture).

metres,
h! - h"
(corrected
for tern-

temperature
of the bath
and for
atmospheric

t.

perature).

pressure
were made.

o

o

o

millims.

millims. o

millims.

millims.

4th curve.

—

4-42

0-06

o-ll

—

0-08

1-19

51

—

4'48

-0-26

-0-47

—

0-12

1-95

52

—

4-22

0-13

0-23

—

0-11

1-64

53

—

4-35

0

0

—

0-09

1-29

54

—

4-35

0-05

0-09

—

0-09

1-18

55

21-7

4-40

0-06

0-11

—

0-09

1 -20

56

—

4-46

-o-io

-0-18

763-9 (22-4)

0-04

0-86

57

—

4-36

0-11

0-20

—

0-06

1-46

58

—

4-47

-o-io

-0 18

—

0-06

1-28

59

—

'4 ■ 37

0-06

0-11

—

0-06

1-27

60

—

4-43

-0-07

-0-13

—

0-06

1-13

61

—

4-36

0-20

0-36

—

0-06

1-32

62

4-56

63

—

—

-0-06

-0-11

—

0-13

2-12

5th curve.

—

4-50

0-13

0-23

—

0-06

1-09

64

—

4-37

0

0

—

0-06

1-06

65

—

4-37

0-18

0-33

—

0-06

0-99

66

—

4-55

-0-13

-0-23

—

0’06

1-03

67

—

4-42

0-03

0-05

763-1 (22-12)

0-06

0-91

68

4-45

69

376

DR. MEYER AY TL HERMAN ON CHEMICAL DYNAMICS

If we now draw curves, taking the times t s as abscissae and the corresponding

i //

7T — 7T

amount of carbonyl chloride formed (i.e., the 7r’s) as ordinates, then the - - ’s or

r2 ~ T1
cItt

- ’s give the rate of formation of carbonyl chloride, or the rate of combination of
dr ^ J

chlorine and carbon monoxide. The curves appear to be remarkably regular, especially
those obtained with the glass bulb. The total number of direct observations is in
Table I. about 70. To trace the nature of the curve through its whole length
observations were made at small intervals, thus dispensing with interpolating results.
Errors arising from the variations of the temperature of the bath, from the variations
of the barometric pressure, &c., can never be completely eliminated by the application
of corrections. For this reason they are greater in the results obtained for small
intervals than when greater ones are taken. By this method the phenomenon is
nevertheless more thoroughly known and its nature more evident, since such an
investigation of the curve does not permit of phenomena characteristic of only one part
of the curve obscuring the true nature of other parts of the curve. As will be seen
from the tables given below this course proved to be necessary in our case, since at
the beginning of the curves we always met with a peculiar phenomenon, called
“ induction,” not characteristic of the rest of the curve.

dir ,

When the

7 r

7 r

To — t, dr

s or — ’s are successively taken on the curve and compared with

one another, we find that they start with very small values approaching zero (the
curve starts asymptotically to the abscissa), and gradually increase till they reach
a maximum, after which they gradually decrease. If we consider curves (l), (2), (3),
(4) and (5) of Table 1. as parts of the same curve, belonging all to one system, we find

dir

that the — ’s gradually diminish, approaching the value of zero, i.e., when no more

(IT

reaction takes place. This takes place when one of the combining substances com¬
pletely disappears from the gas mixture.

• . dir • . •

An investigation of the curves, after the — ’s arrived almost at their maximum,
° dr

shows with absolute certainty that the equation

[log, (A — aq) — log, (A — x2) -f log, (B — x.2) — log, (B — aq)] : (r.2 — iq) = C (l)

(a constant) holds good, where A and B are the quantities or volumes or partial
pressures of chlorine and carbon monoxide before the reaction was first started,
expressed in millimetre pressure of the manometer. A — aq, A — aq, B — aq, B — aq,
are the quantities of chlorine and of carbon monoxide present in the system at the
times t : and r2 (see Table II. below).

It is thus evident that our integral equation must be

JZTb P°& (A ~ ~ lo& (A _ xi) + lo& (B — xi) - log‘- (B - *])] : (t2 - b) = K (2).

AND STATICS UNDER THE INFLUENCE OF LIGHT.

377

The differential equation giving the law of velocity of reaction is thus

f = K (A - z) (B - *) . (3),

%. e. , the velocity of combination of chlorine and carbon monoxide in light, or the
velocity of the formation of carbonyl chloride, is at the time r directly proportional to
the product of the reacting masses at the time r. Since the chemical equation for
the reaction is Cl2 + CO = CO Cl2, this equation, in light has the form which it ought
to have according to the law of mass action in homogeneous systems, if the chemical
reaction were to go on in the dark as the outcome of those intrinsic properties of
matter only, which we call chemical affinity or chemical potential. In the above
equations K is the velocity constant, which gives the velocity of combination of chlorine
and carbon monoxide under given conditions of experiment, when A — x for chlorine
is 1 and B — x for carbon monoxide is 1. K in the above equations is evidently also
an integral velocity constant for all wave lengths of the acetylene light, the value of
(K) being different for each wave length. Since, however, each wave length has an

Jx

equation of the same form — = (K) (A — x) (B — x), the equation for light consisting

of more than one wave length remains the same ; K or (K) is besides a function of the
intensity of light, of the temperature, and of the surrounding medium.

In the following tables (II., III., IV. and V.) :

No. is the number of the observation.

tt' — n" = dx is the amount of carbonyl chloride formed, or of chlorine or of carbon
monoxide which has disappeared during the times r2 — r1.

A — x is the quantity of chlorine present in the system at the time r.

B — x is the quantity of carbon monoxide present in the system at the time r.
r2 — is the time between two successive observations.

Equation (2) should be true if the law of mass action holds good.

— gives the rate of formation of carbonyl chloride at the successive times ; this

ought to be constant, if the rate of formation of carbonyl chloride in a unit
of time were independent of the reacting masses and were directly proportional

to the intensity of the light introduced only.

St

i;(A — x ) gives the rate of formation of carbonyl chloride at the time r, divided

by the quantity of chlorine present in the system at the time r. This ought
to be constant if the rate of formation of carbonyl chloride at a given intensity
of light were directly proportional to the amount of light absorbed by the
system (i.e., by chlorine) in the unit of time during the reaction.

3 c

VOL. CXCIX. - A,

Table II. — Cl3 and CO (in the Glass Bulb). Experiment started 5th July, 1901. Temperature ca. 210,

378

DR. MEYER WILDERMAN ON CHEMICAL DYNAMICS

N

•

05

XH

co

ZD

rH

o

b

05

rH

Xh

co

CM

CM

rH

CM

o

o

co

o

05

CO

c

to

CM

CM

to

CM

to

Xh

M

CO

o

o

o

o

o

iH

rH

CM

<M

r— H

CM

05

jH

CM

CM

CM

<M

<M

r"1

CM

1

■S3

o

rH

o

CM

00

b

CM

IH

CO

05

CM

05

CO

Xh

CM

x>-

<M

H

CM

X^

to

o

05

to

X>»

CO

05

CM

rH

CM

to

b

b

b

CO

-e

x^

C

o

i— H

b

o

H

co

iH

CM

o

co

o

o

oo

b

r-

o

o

o

rH

00

CM

to

b

CM

CO

GO

O

**

H

CM

CM

CM

CM

(M

rH

CM

X •

o

W <M

05

to

1H

iH

rH

CM

CO

CM

to

x>-

b

GO

co

CO

X-

b

xh

b

pq v1

CO

O

o

CO

GO

co

GO

CO

GO

co

CO

to

bJD

l N

O

1

o

o

o

o

o

CM

b

GO

cc

GO

GO

GO

x^

GO

n

C~l

b

n r i

fc*

co

to

to

to

2

tO

to

o

o

to

o

05

o

o

05

o

o

o

o

Cl

PS

rH

b

*H

oo

co

iH

rH

H

iH

rH

rH

rH

rH

b

a

H

1 - l^r-n

II . 1 ?

X—

CO

CM

o

to

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AND STATICS UNDER THE INFLUENCE OF LIGHT.

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386

DR. MEYER WILDERMAN ON CHEMICAL DYNAMICS

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AND STATICS UNDER THE INFLUENCE OF LIGHT.

387

3 D

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388

DR. MEYER WILDERMAN ON CHEMICAL DYNAMICS

In the above tables curves (1), (2), (3), (4), and (5), all belong to one system
at 21°-7.

Che values of K in the above tables, calculated from equation (1), show, that as
the system is brought from the dark to the light the reaction at first does not appear
to be going on, then it goes on very slowly giving very small values for K, gradually
the values for Iv, and with it the speed of reaction, increase until the values obtained
for K remain constant, the speed diminishing at the same time according to the law

AND STATICS UNDER T1IE INFLUENCE OF LIGHT.

389

of mass action. If after the values of K remain constant for a sufficient time, the
light is removed for some hours, and the system again exposed to the light, we again
find the same phenomenon ; the velocity constant K is not obtained at once but only
after the reaction has gone on for some time. Further, it is found that in every
case the same velocity constant is obtained after a time. It follows from this that
the combining chlorine and carbon monoxide when exposed to light of a certain
intensity and composition, always acquire after a certain time the same constant
properties, the same chemical affinity to one another. The fact that the same
constant was found in all curves of the same system, and that the investigation was
carried on as far as 80 '66 per cent, (in Tables I. and II.) and 39 T 6 per cent, (in
Table III.) of the total amount of possible combination, shows that the above
equation (3), p. 377, which is a true expression of the law of mass action, truly
represents the fundamental law underlying chemical kinetics in light. At the same
time the last two columns of the above tables illustrate beyond any doubt that it is
no longer possible to assume that a law analogous to Faraday’s for electrolysis
governs the phenomena of chemical kinetics in light.

Instead of getting a constant in the last columns, the values of dx/dr- 100 fall
from 2 5 '2 to 4T (in Table II.), and from 40 to 18 '6 (in Table III.), and the values of

~ : ^ ^ (in Table II.) fall from 50'9 to 9'9, and (in Table III.) from 205

to 136. Special attention should be given to the curves (l), (2), (3), (4), and (5),
of Table II. Here a large quantity of chlorine and a small quantity of carbon
monoxide were employed ; in this way the variations in the quantity of carbon
monoxide were increased, and in that of chlorine made small, i.e., it is the variation
in the quantity of the carbon monoxide which absorbs but little light, and not in the
quantity of the chlorine, which absorbs much, which in this case proves to be the
main cause of the velocity of reaction decreasing so rapidly. It is thus evident that
it is not the quantity of light absorbed by the molecules in the unit of time, but the
quantity of the reacting substances present, which determines the velocity of the
reaction, no matter what quantity of light the molecules absorb, provided that under
the action of light the atoms and molecules acquire that quantity of energy which is
characteristic of them after the period of “ induction ” has passed. In other words a
system containing two molecules chlorine and one molecule carbon monoxide will
combine at the same rate as a system containing one molecule chlorine and two
molecules carbon monoxide, though the first system absorbs almost twice as much
light as the second.

The “ Induction ” and “ Deduction ” Periods of Energy of the System and the
Chemical Periods of “ Induction ” and “ Deduction ” in Light.

Having considered those parts of the curves where the velocity constant can be
traced, we now consider the parts before the velocity constant is reached. We find

390

DR. MEYER WILDERMAN ON CHEMICAL DYNAMICS

that, starting with a system such as chlorine and carbon monoxide, at first it seems
for some time that no combination takes place at all. Combination, however, becomes
gradually more and more apparent, the velocity becomes greater and greater, till a
constant value for K is obtained. This is a peculiar phenomenon, which Buxsex
and Pv-OSCOE first observed in the case of chlorine and hydrogen, and which they
appropriately called the “ induction period. If now, after the velocity constant has
been observed for some time, the light is removed from the system, the reaction at
once becomes very slow and soon stops. If the system is, after a time, again exposed
to light, we find at first an “ induction period,” after which the same velocity
constant is obtained. This shows that as the light is removed from the system the
reacting molecules lose the properties which they had acquired in the light, gradually
returning to their old state ; and that when the system is again exposed to the light
the molecules and atoms each time gradually acquire the same new properties.
Besides the “ induction period” we thus have to deal also with a “ deduction period.”
The “ induction period” is evidently not due- to the absence of some product of the
reaction, but is a period during which the molecules and atoms of the systems
continuously change their state of energy from that in the dark to that in the light ;
and the “deduction period” is a period during which the molecules and atoms
gradually return from their state of energy in the light to the state of energy they
possess in the dark. The properties of the “induction” and “deduction” periods
require, however, still further consideration. On removing the light from the system,
that state of energy of the atoms and molecules which makes them capable of
entering into reaction rapidly passes away, with the energy stored in the molecules
and atoms under the action of light, and is transformed partly into heat and partly
in chemical action as long as this goes on after the removal of the light.

The curves given above show, however, that while chemical action ceases, or
apparently ceases, after a short time, it takes a considerable time before the atoms
and molecules again completely acquire the properties which they previously had
in the dark. Thus, between the first curve and the second (between 7 and 8) the
light was removed for 16 hours and 48 minutes, between the third and fourth
(between 47 and 48) for 17 hours, between the fourth and fifth (63 and 64) the light
was interrupted only for 5 minutes, and the rest of the time (39 minutes) it was
again exposed to the light. We nevertheless find that, after 17 hours, the “induction
period ” of the second curve did not again become as slow as it was in the first
curve; after the third interruption of the light, for about the same 17 hours, the
“induction period” was almost quite the same — starting with almost the same
values of K ; again, during the fourth interruption of the light for only a few
minutes, the chlorine and carbon monoxide of the system returned only so little
to the properties which they had at first in the dark that no marked variation
in the value of the velocity constant K could be established. Thus we find a
remarkable analogy (though not a reversible identity) between the period of “ deduc-

AND STATICS UNDER THE INFLUENCE OF LIGHT.

391

tion ” and “induction.” On exposing the system for the first time to light, the
energy of the light is absorbed by the system for a considerable time before the
atoms and molecules acquire the state of energy when the reaction becomes apparent,
i. e. , the energy of the system first continuously increases and changes though no
chemical effect can be perceived in the system. When the chemical reaction
becomes evident the chemical “ induction period” then continues till the new constant
state of energy characterised by K is reached. On removing the light, the chemical
“deduction period” continues only a short time, when the reaction ceases to take
place, but it takes a long time for the atoms and molecules to lose those properties
which they acquired in the light before chemical reaction started, and this gradual
diminution of the energy of the system is again not to be discovered from the
chemical reactions of the systems. The chemical “ deduction period,” however, lasts
much less time than the chemical “induction period.” It is evident that the curves
of the “ induction ” and “ deduction periods ” given above only rejmesent the amount
of chemical transformation, i.e., are curves of chemical “ induction ” and “ deduction,”
and are not the curves which represent the gradual increase and decrease, the
“induction” and “deduction” periods, of the whole energy of the system, when it is
exposed to light or when light is removed from the same. There are other methods
by which the variation of energy during the “ induction ” and “ deduction ” periods
may be determined. The author is now engaged in the elucidation of the laws
concerning the induction and deduction periods of energy which up to the present
have only the character of qualitative observations.*

The InjliLence of Small Traces of Air and Water upon the Mixture of Pure
Chlorine and Carhon Monoxide ( and other Gaseous Systems ).

The admixture of small traces of air with the reacting gases jnoduces a most
remarkable retarding effect upon the velocity of the reaction. Bunsen and Boscoe
found that this held for a mixture of chlorine and hydrogen, so that it appears to
be a general rule for all gaseous systems.

Two “ quartz vessels ” were placed one behind the other, both vessels were
evacuated and treated in exactly the same way, and finally filled with carbon
monoxide from the same sample at the same time. Chlorine from the same sample
was driven by means of the same concentrated sulphuric acid (which, for reasons
mentioned before, it was impossible to keep for any length of time quite free from

* As to the chemical induction and deduction periods it is evident that, since velocity of reaction
follows the law of action of mass, when the molecules taking part in the reaction have attained, under
the influence of light, a constant value of their chemical potentials, the same law of mass action must also
he the governing principle for the velocity of reaction at any given moment of the chemical induction and
deduction periods, only the velocity constant, K in equation (3), will vary in time as the chemical
potentials of the reacting substances change.

392

DR. MEYER WILDERMAN ON CHEMICAL DYNAMICS

air) first into the vessel behind, and then, about a quarter of an hour later, into the
vessel in front. Both vessels were simultaneously exposed in the bath to the same
acetylene light, but the velocity of combination of chlorine and carbon monoxide was
considerably slower in the front vessel than in the vessel placed behind it. (The
opposite would have been expected.) (See Curves 1 and 2 of Tables III. and IV.)

The accelerating influence of water vapour upon the velocity of reaction is very
great. The same glass bulb was filled with the chlorine and carbon monoxide, both
freshly prepared in the dark. The chlorine was in both cases absolutely the same,
i.e., equally dry; the freshly prepared carbon monoxide on the contrary was allowed
to bubble in the two experiments through different heights of sulphuric acid in the
bulb S (fig. 1, p. 340) by tilting the bulb more in the second experiment than in the
first. The sulphuric acid was the same in both cases, and each time was first heated
in the bulb in a vacuum till it was perfectly freed from any gas. The freshly-prepared
carbon monoxide was thus most probably in one case a little drier than in the other,
and the difference in the quantity of water still retained by the carbon monoxide
evidently could only be exceedingly very small. The difference in the speed ol
combination of carbon monoxide and chlorine on the contrary proved to be very great.
(See curves of Table V.)

A further experiment was thus made : — The gases were brought into the glass
bulb containing ordinary concentrated sulphuric acid freed from air, well shaken, then
left in the dark for about two days, and then exposed to the powerful acetylene
light ; no reaction could be observed after several hours during two days, but
on exposure to sunlight the two gases combined, though only exceedingly slowly.
Since the concentrated sulphuric acid was of sp. gr. P84, it contained water, and it
must still have had (Regnault) a vapour pressure of water, though an extra¬
ordinarily small one, so that the pressure of water vapour was still not absolutely
excluded, and this may be the cause why an exceedingly slow reaction could still take
place in sunlight.

Thus, the less water vapour is present with the gases the slower is the reaction.
Reaction takes place in the presence of an exceedingly small quantity of water vapour
in the mixture ; small, apparently immeasureable, differences in the amount of water
vapour, at any rate when the vapour is present only in small quantities, produces
great differences in the velocity of the reaction.

It should be observed that we could not use greater quantities of water vapour for
the experiments, still less could we have the gases in presence of water, because water
decomposes the carbonyl chloride formed, giving carbonic and hydrochloric acids.

Bunsen and Roscoe used their mixture of chlorine and hydrogen in the presence
of water saturated with these gases. Piiingsheim found later on that, if chlorine
and hydrogen are taken quite dry, they do not combine in light at all. This
phenomenon Pringsheim thought possible to explain in the following manner : —
The chlorine and hydrogen, according to his conception, do not combine directly to

AND STATICS UNDER THE INFLUENCE OF LIGHT.

893

hydrochloric acid, but chlorine and water first form an intermediate compound
hypochlorous acid (and hydrogen), which with hydrogen forms the hydrochloric acid,
setting free the same molecules of water which are again used in the reaction, i.e.,
instead of having H2 -f- C1.3 = 2HCL, we have Cl2 + H20 = CLO -f H3 ; CLO + H4
= 2HC1 + HoO. I doubt the correctness of this explanation for the following
reason : — It is known that chlorine and water, when exposed to light, form not
hypochlorous acid but hydrochloric acid (Wittwer, Bunsen, and Roscoe).^ Still
the difficulty remains that chlorine and hydrogen are not the only system having
such properties. The system chlorine and carbon monoxide exhibits the same
property. We also know that ammonia and hydrochloric acid do not combine
when perfectly dry ; carbon monoxide and hydrogen do not explode when perfectly
dry (Dixon) ; perfectly dry hydrogen peroxide does not act upon a photographic
plate (Russell). For this reason it seems that in gaseous systems a phenomenon is
met which in other cases is called “catalytic action,'’ in which a reaction is accelerated
or caused by the presence of an extraneous substance, which apparently or in reality
takes no part in the reaction, e.g., the action of platinum-black upon the decompo¬
sition of hydrogen peroxide, &c.

The catalytic action of gases upon the velocity of reaction in the gaseous systems
may be divided into “catalytic action with an accelerating influence” (to this belongs
the action of water vapour upon the gaseous systems mentioned above) and into
“ catalytic action with a retarding influence” (to this belongs the action of oxygen or
air upon the systems chlorine and hydrogen or chlorine and carbon monoxide). This
division is, however, of a purely formal nature, and hardly anything is known of the
ultimate nature of the phenomenon.

Velocity of Chemical Reaction and Chemical Equilibrium in Light.

Having thus established beyond any doubt that the velocity of reaction in light is
governed in homogeneous systems by the same law of mass action as in the dark, the
influence of temperature as well as of the intensity of light upon the value of the
velocity constant, as well as the connexion between the velocity constant and the
wave length on the monochromatic light, have still to be investigated ; and finally
the investigation of heterogeneous systems regulated by other fundamental principles
remains necessary. This will form the subject of the author’s future investigation.

Ou the other hand the solution of the problem for chemical kinetics in light
evidently already shows with perfect certainty that the law of mass action, which
governs chemical equilibrium in homogeneous systems in the dark must necessarily
also govern chemical equilibrium in the light. It is most remarkable that we do not

* Professor Dixon, in a private communication, makes a better suggestion — that the reaction between
hydrogen and chlorine is to be conceived thus : — CL + ILO = 2HC1 + 0 (nascent) ; O l- LL = Hl.O,
the first equation being reversible.

VOL. CXCIX. - A. 3 E

394

DR. MEYER WILDERMAN ON CHEMICAL DYNAMICS

know of any reversible system, in which the two opposite reactions do not go on in
the dark but go on in the light. But we do know systems in which one reaction
goes on in the dark and is modified in light, while the opposite reaction goes on in
the light only, e.g .,

2AgCl in solution ! _ ; Ag0 (or Ag,Cl) in sol. -f- Cla in sol.

it 'it

AgCl solid Ag (or AgoCl) solid.

Here is an homogeneous system, on the one hand silver chloride is decomposed by
light into silver and chlorine or silver sub-chloride and chlorine (this question is at
present unsettled), on the other hand silver (or silver sub-chloride) and chlorine
combine in the dark forming silver chloride, and this combination evidently goes on
in light also (though probably with a different speed). Let the volume of the solution
be Y. For the first reaction in light we have according to the law found above

dx (A -
~T — c - V

CIT V

_ /JQ

— , wdrere - is the concentration of the molecules of silver chloride in

solution (however small this may be) at the time r, and x/v is the concentration of
the chlorine as well as of the silver molecules formed in solution ; c is the velocity
constant which changes with the intensity and composition of the light passing
through the system. For the second reaction we have according to the law of mass

, where c is the velocity constant for the reaction in

which silver chloride is formed from silver and chlorine. The velocity constant in
light is different from that in the dark, say c", however small this difference may be,
for the reason that chlorine and silver are in a different state of energy in the dark
and in the light. It follows from this that when equilibrium takes place in the light,
or when no further variation in the masses takes place,

action in the dark ( — ) = c' -

dx

A /

dr

x

V

dx _ (ch:d
t It \dr)

= 0

c (A - A2

(A - xy

c

that is, we must at the point of equilibrium get a constant K, which will regulate the
masses forming the reversible system with the variation of the volume or of the
concentrations or of the partial pressures of the substances, because both opposite
reactions have each a separate velocity constant before equilibrium. Though this
proof of the necessity of the existence of a constant of equilibrium is absolute, it
would have been very valuable and desirable to directly illustrate the equilibrium
constant K in a reversible system from the varying masses at equilibrium, as we
succeeded in doing for the velocity of reaction. Unfortunately there is not one
homogeneous system known where such a proof could be successfully carried out. It
is well to remember the enormous difficulties one meets with in this region, when even
such apparently simple reactions as the combination of carbon monoxide and chlorine

395

AND STATICS UNDER THE INFLUENCE OF LIGHT.

or of chlorine and hydrogen, are to be measured quantitatively. The author is at
present engaged in such attempts to test directly the constants of equilibrium, not so
much because the law needs further confirmation, as on account of the very interesting
thermodynamic connection which must exist between the constant of equilibrium, the
heat of reaction in light, and the absolute temperature on the one hand, and the
constant of equilibrium and the intensity of light on the other.

Appendix : Thermodynamical Considerations .

The above experimental results find their rational basis and explanation in thermo¬
dynamics. The condition of equilibrium in a homogeneous system, when oidy
chemical, thermal, and mechanical energy are taken into consideration, is according
to Gibbs: c/E = t dy — pdv + yldm1 + p,2c?m3 . . . + \i,ldmn 0, where E is the
energy, y the entropy, ml5 m.2 . . . mn the quantities of the substances Sl5 S2 . . .,
the chemical potentials of Sl5 S3 . . . Let us now assume that the system is exposed
to light of constant intensity and composition, and that the system is in such thin
layer that the intensity of the light is the same in all parts of it. Since all
substances absorb light and the light absorbed is not completely transformed into
heat, a part of the light will appear as other forms of kinetic energy of the atoms
and molecules. From a molecular mechanical point of view this will mean that
under the influence of light the amount of work present in the molecules as energy
of the atoms increases. Obviously the ratio of the amount of light transformed into
heat to that transformed into kinetic energy of the atoms is not constant. At first
the energy of the atoms and molecules gradually increases (induction period of
energy), until a reaction, a shifting of the point of equilibrium to another one, becomes
possible (chemical induction period), which is observed by an increase of the velocity
constant. Under the action of light the storage of energy in the atoms and molecules
ultimately reaches a maximum, after which light produces no more strain upon the
atoms, preventing them only from losing the energy once acquired, and the whole of
the light entering into the system is transformed into heat. This maximum kinetic
energy of the atoms is a function of the intensity and composition of light, of the
nature of the substance, of the surrounding medium, &c., and becomes apparent in
the fact that a velocity constant, indicative of constant properties of the atoms and
molecules, is obtained. When light is removed from the system the energy stored in
the atoms and molecules, under the impulses of the light waves, gradually disappears,
changing either into chemical energy and heat (as long as the reaction continues in
the dark, chemical deduction period) or into heat alone (deduction period of energy).
When the maximum kinetic energy of the atoms is reached under the action of light,
it is evident that the energy stored up must be directly proportional to the mass of
each substance. To the above equation for equilibrium the terms vldml + y,cZm2 . . .
+ v,ldm)l must therefore be added. By means of a cycle process at a constant

396

DR. MEYER WILDERMAN ON CHEMICAL DYNAMICS

temperature it can also be shown that the entropy of the system changes in light.
When chemical transformation takes place in the system this is also accompanied by
variation in its mechanical energy. The kinetic explanation of the phenomena of
absorption, dispersion, fluorescence, by Stokes and Helmholtz, led the author to
the conclusion that the energy stored in the atoms and molecules under the action of
light, partly transforms into chemical, partly into kinetic energy sui generis, which
may be called light -kinetic energy, — a conclusion strengthened by the author’s
experiments on the effect of light upon two plates of the same element, when they
are immersed in a liquid, connected with a galvanometer, and one plate is exposed
to light while the other is kept in the dark. Thus, under the action of light, the
chemical potential of each substance increases and each substance acquires a new
light-kinetic potential. Instead of equation (i.) we now have for equilibrium in light

c/E + c?Ej = dE' = t' dr)' — p'dv ' + y / dm{ -f- p2dm2 . . . -j- \in dmj

+ dm y -j- \2dm2 . . . -f Xn' dm J ^ 0 . (ii.).

Integrating this equation, then differentiating in the most general way and
subtracting (ii.) we get

vj'dt ' — v'dp' -j- m'dky -f m'd/iy . . . Jr mjdkj -j- mjdpj = 0 . . (hi.).

General considerations show that for the system to be in equilibrium the sum
of both potentials of each substance must be constant through the whole system,
i.e., /a/-k X1'=c1, p.2' + \2 — c2 . . . (y). (iii.) and (y) give the variation of temperature
or pressure, or of the chemical potential, or of the light-kinetic potential, or of several
of them, with the variation of one or more of the rest of the variables. The sum of

GE + f/E

both potentials /x/ + X/ being =

clrtiy

1 ' and the equation for chemical

reaction being tqAj + n2 A2 = n3A3 (a), we still find that, under due considerations,

ni (hi + X/) + % (hz X/) = n3 (p3/ + X3') . (/3).

Taking in equation (iii.) the grammolecule as unit of mass (which is not the case in
Gibbs’ deductions), in order to get subsequently a result which in its form and
content expresses our present molecular conceptions of a chemical reaction, &c., we
get, if the system is a gaseous one, consisting of one substance only, and its total mass
is grammolecules, that the total chemical energy is p/m/, the variation in the
same m/d/x/, the total light-kinetic energy is X/m/, the variation in the same ?n1/dX1/,
the total mechanical energy p’v = m/RT, since pv of 1 grammolecule = ET, and

v dp'= v'd

rt/RT

'H'

/

the total entropy of the mass pr—my ( — -j-K/ ), when the entropy

of 1 grammolecule =

II ' (of 1 grammolecule)

-f- K'. Thus putting in (iii.) these values

m '

and integrating we get /q -f ,\1 = RT -f RT log — ~ — IT log T K' T + K", where

v

AND STATICS UNDER THE INFLUENCE OF LIGHT.

3 9?

K; and K" are integration constants. Assuming now that the system consists of
several substances, as given in (a), and that the law of Dalton holds good for the
chemical and for the light-kinetic potentials, then we get from (/3) that

io£r { a-y ■ (^r/m = sr ikk.' - t - (»» + »i - %) bt

+ (niH,' + njH,' - «3H3') log T + («3K3" - K" - %Ka")] . . (iv.),

where —7 5 Dl , Ul are the concentrations of each substance expressed in gram-

m.

m,

v

molecules, nx, n2 . . . the numbers of grammolecules of each substance taking part in
the reaction, u<3K3' — nx K/ — n^KJ — constant EA, — (n3+% — w3) RT is the work done
by the system (a) during the transformation in light, (?qH/ -j- njl 2' — n3 H3') log T
+ (n 3^3" - «iK 1" — is the heat of reaction in light. Thus the connection

between the logarithm of the constant of chemical equilibrium in homogeneous
systems in light, the heat of reaction or of transformation of ?q grammolecules
of Sj plus n2 grammolecules of S2 into n?j grammolecules of S3 in light, the work done
during the transformation, and the absolute temperature, follows the same law in light
as it does in the dark. The effect of light upon a system therefore consists in shifting
it to a new point of equilibrium. It is further easy to show that at a constant
volume, since the work — (??2 + nx — n3) B/T = 0, and r/t' can be put — . C,T, where
C, is the specific heat at a constant volume, T the absolute temperature, an equation

is arrived at, which after differentiation gives

lQg(v

~k 1°

P*

fpf'

i

& \ V’

n2

A + BT
RT2

,(v.), where A and B are constants (equation of Uan’t Hoff and Kooy),

’m2'\n‘ l fm2' \r‘3

which at a constant temperature gives [prj ’ j ~ constant (vi.), i.e., the

law of mass action holds good for chemical equilibrium in light, as found experimentally.
Decomposing this equation for homogeneous systems in the usual manner into two,
giving the two opposite velocities of reaction, which at equilibrium become equal,
we get

/ _7 - . ' / f ... / , v,. / /\ -.j. / .7 \ // I ' \

. . (vii.),

/ dec V

( dT ) ~C \v

/ / "ll

m/\n' /md\n*

yra„dUU" = c"(^r.

i.e. , the velocity of chemical reaction in light must also follow the laws of mass
action, as found experimentally.

In conclusion, I should like to express my thanks to the Managers of the Davy-
Faraday Laboratory of the Royal Institution for having allowed me to make use of
the splendid arrangements of the Laboratory, and especially to Dr. Ludwig Mond,
who by his kind assistance has enabled me to undertake and carry out the above
research, and who by his valuable advice on many occasions has very essentially
contributed to the success of the same.

INDEX SLIP.

Dpnstan, W. R., and Henby, Thos. A. — -Cyanogenesis in Plants. Part II. —
The Great Millet, Sorghum vulgare.

Phil. Trans., A. yoI. 199. 1902, pp. 399-110.

H enby, Thos. A., and Dttnstan, W. R. — Cjanogenesis in Plants. Part II. —
The G-reat Millet, Sorghum vulgare.

Phil. Trans., A, vol. 199, 1902, pp. 399-410.

Dhurrin, Constitution and Reactions of — Formation of Dhurrinic Acid from.
Dttnstan, W. R., and Henby, T. A.

Phil. Trans., A, vol. 199, 1902, pp. 399-410.

Hydrocyanic Acid, Formation of, from Sorghum vulgare.

Dunstan, W. R., and Henry, T. A.

Phil. Trans., A, vol. 199, 1902, pp. 399-410.

Parahydroxymandelic Aeid, Preparation of.

Dttnstan, W. R., and Henry, T. A.

Phil. Trans., A, vol. 199, 1902, pp. 399-410.

Sorghum vulgare, Glucoside and Enzyme of.

DrNSTAN, W. R., and Henry, T. A.

Phil. Trans., A, vol. 199, 1902, pp. 399-410.

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[ 399 ]

VII. Cyanogenesis in Plants. — Part II. The Great Millet , Sorghum vulgare.

By Wyndi-tam It. Dtjnstan, M.A., F.R.S., Director of the Scientific and Technical
Department of the Imperial Institute , and Thomas A. Henry, D.Sc. Lond.

Received April 24, — Read May 15, 1902.

In a previous paper, our first communication on this subject (‘ Phil. Trans.,’ B,
vol. 194, 1901, p. 515), we have shown that the poisonous effects produced by the
young plants of Lotus arabicus are due to prussic acid, which is not present in the
plant as such, but originates in the hydrolytic action of an enzyme, lotase, on a
glucoside, lotusin. Recently we have examined a large number of plants which, like
this Egyptian vetch, appear, under certain conditions, to possess poisonous properties,
and at other times to be innocuous and often valuable as fodder plants or food stuffs,
with the view of ascertaining to what extent they contain glucosides furnishing
prussic acid.

Among the first of these plants we examined was the Great Millet, Sorghum
vulgare, a plant widely cultivated in tropical countries for the sake of its nutritious
grain, which in many districts of India is the staple food, known as “ Juar,” of the
natives. In the West Indies what is apparently the same plant yields the important
“ Guinea Corn” and in South Africa “ Kaffir Corn.”

We were informed by Mr. E. A. Floyer, of Cairo, that in Egypt it is well known
to the Arabs that the green portions of the young plant — the vernacular name of
which is “ I) hurra Shirshabi” — are poisonous, and that during this period the
plantations are protected in various ways in order to prevent cattle from feeding on
the immature growth. It is to be noted that in Egypt the name “dhurra” is also
applied to a variety of maize which is largely cultivated.

Mr. Floyer has given us the following account of the plant in Egypt. “ Dhurra
shirshabi ” is not grown in Egypt as a crop, the yield of corn being too small. It is
planted chiefly in order to shade the “ Arachis” (ground nut), to which it also affords
protection in forming a poisonous hedge. The “ thinnings” of the young Millet are
often strewn around a cultivated crop, and the neighbours are warned to keep their
cattle off. The poison is most intense when young plants, 1 foot high or less, are
kept without water for a long time, and such unwatered young plant is highly toxic
to cows. The plant appears to have been brought to Egypt from Syria, and is now
grown chiefly at Bir Abu Bala, near Ismailia. The “ fellaheen ” do not plant it.

(318.) 24.10,02

400

PROFESSOR WYNDHAM R. DUNSTAN AND DR. THOMAS A. HENRY

Cases of poisoning by young Sorghum have been also recorded in America and in
Australia, where the plant is grown for forage purposes.

In India the poisonous properties of the plant — which bears the vernacular name
“juar” or “jowar” — do not appear to be so generally known, although several
well authenticated cases of the poisoning of cattle by it, especially during drought,
have been recorded, and much has been written on the subject by veterinary surgeons
and others, who have, as a rule, assumed that the toxicity is due to the presence of
a poisonous fungus or insect upon the plant, or that the Great Millet is not naturally
poisonous, and that the deaths of cattle as the result of eating it are due to
immoderate consumption, which causes a kind of suffocation from indigestion,
technically known as “ ho veil.” The symptoms of “ hoven” are not unlike those of
prussic acid poisoning, and it is possible that the various leguminous fodders which
are known to be particularly liable to produce these effects may, at any rate in
some cases, prove, like Lotus arahicus, and, as will be shown in the present paper,
Sorghum vulgare, to furnish prussic acid.

For the material we have employed in the course of this investigation we are
indebted to Mr. E. A. Floyer, who was good enough to undertake its collection in
Egypt at different stages of growth.

Considerable confusion exists as to the identity of the “ Great Millets” grown in
different tropical countries. Thus in India the plant is cultivated both as a spring
and an autumn crop. The varieties ripening in the spring are probably originally
derived from Sorghum halapense, a species indigenous to India, whilst the autumn
crops are generally referred to Sorghum vulgare, yet both spring and autumn crops
are called “ juar” or “jowar,” and are used by the natives indiscriminately. Again,
in India a plant with an inflorescence more branched than that of Sorghum vulgare has
been regarded as a distinct species, and named Sorghum saecharatum ; this name is
however given in the ‘ Index Kewensis’ as a synonym for Sorghum vulgare, of which
the plant is probably merely a variety.

The plant we have examined has been identified for us by Dr. Schweinfurth as
undoubtedly true S. vulgare.

Preliminary Experim en ts.

It was observed that the young plant when crushed and moistened with cold
water soon acquired a strong odour of hydrocyanic acid. The production of this
acid was confirmed by pressing out a little of the liquid from the moist plant, and
distilling it, when a liquid was obtained which gave the characteristic reactions of
hydrogen cyanide.

A few grammes of the plant were next exhausted by hot methylated alcohol in a
Soxhlet extractor. The solvent was distilled from the solution and the residue
boiled with water until nothing more dissolved. The aqueous liquid was then

ON CYAN OGENESIS IN PLANTS.

401

distilled at first alone, and afterwards with the addition of dilute hydrochloric acid ;
in the former case none, but in the second, where hydrolysis had occurred, con¬
siderable quantities of hydrocyanic acid were found in the distillate.

These observations led us to conclude that Sorghum vulgare contains a glucoside
which under the influence of some hydrolytic agent simultaneously present under¬
goes hydrolysis, furnishing as one product hydrocyanic acid, to which the observed
toxicity of the young plants must be ascribed.

A determination of the amount of acid which the air-dried plant is capable of
producing at different stages of growth was made by leaving a weighed quantity
in contact with water for 12 hours, and distilling off the acid formed in a slow
current of steam, the liquid being titrated by LiebicLs method.

The following results were obtained ; —

C5

( а ) From bright green plants about 12 inches in height ;

20 grammes gave a distillate requiring 7 ‘45 cub. centims. ' silver nitrate,
equivalent to ’201 per cent. HON.

1ST

20 grammes gave a distillate requiring 7 ‘8 cub. centims. — - silver nitrate,
equivalent to ‘216 per cent. HON.

(б) From plants about 3 feet high, yellowish-green and ripe ; 20 grammes of
these mature plants gave no indication of prussic acid, and larger quantities on
distillation with water gave amounts too small to be satisfactorily estimated. No
prussic acid was obtained from the seeds of the Millet.

It has Jbeen asserted by Greshoff and Treub that in many tropical plants
hydrocyanic acid occurs as such, that is, in the free state. The existence of the
free acid was demonstrated by these observers by immersing a thin section of the
plant first in alkali, then in a mixture of ferrous and ferric chlorides, and finally in
strong hydrochloric acid. If the plant tissue was stained blue, it was concluded that
prussic acid in the free state was present. This test, however, appears to us to he
quite inconclusive, as the mere moistening of plant tissue containing Loth a glucoside
capable of furnishing prussic acid on hydrolysis and a hydrolytic enzyme, leads to
the immediate production of free acid, which by Greshoff and Treub’s method
would be regarded as occurring pre-formed in the plant. We have carefully
examined various specimens of dhurra for free prussic acid by the following methods.

About 20 grammes of the finely-powdered plant were placed in a distilling flask
attached by its branch tube to a long condenser. Into the closed flask a rapid
current of steam was passed, which served the double purpose of immediately
destroying any enzyme, and of carrying through the condenser any volatile product
present in the plant. In the distillate of the plant thus obtained we never found
prussic acid, either with young Sorghum vulgare or Lotus arabicus.

It therefore appears that, like Lotus arabicus , the poisonous effects of the young

VOL. cxcix. — a. 3 F

402

PROFESSOR WYNDHAM R. DUNS TAN AND DR, THOMAS A. HENRY

<llmrra are due to the presence of a glucoside, which yields prussic acid under the
influence of an enzyme also present in the plant.

Extraction of the Glucoside (Dhurrin).

The finely-powered plant was extracted with alcohol, the solvent distilled off and
the residue warmed with water until nothing more dissolved.

To this liquid aqueous lead acetate was added so long as a precipitate formed.
The precipitate (lead tannate, &c.) was removed. The filtrate, which was now
bright yellow, was treated with sulphuretted hydrogen, care being taken to avoid a
large excess, and the lead sulphide was removed by filtration. A stream of air was
then drawn through the liquid to remove hydrogen sulphide, and the solution
evaporated in a vacuum. After several weeks the syrup deposited a small
quantity of a crystalline substance, and more was obtained by adding small quantities
of alcohol and dissolving the mixture of precipitated sugar and glucoside in a little
water, and setting aside to crystallise as before. This process was very tedious, and
the two following methods have been since found to yield the glucoside much more
rapidly.

A. The liquid, after the hydrogen sulphide treatment, is evaporated in a vacuum to
a convenient volume, and the amount of free sugar determined with Fehling’s
solution. A little more than the calculated quantity of phenylhydrazine necessary
to convert this amount of sugar into the osazone is then added, and the mixture
heated for 30 minutes at 100'J C., filtered, and the filtrate shaken with ether to remove
any excess of phenylhydrazine. On evaporation in a vacuum the residue generally
solidified to a mass of crystals, which were easily purified by recrystallisation from
alcohol. The method always involves the loss of some of the glucoside, and cannot
be employed in the isolation of small quantities.

B. The second method, which is the more effective, consists in evaporating in a
vacuum the extract left after the lead acetate and hydrogen sulphide treatment
with sufficient purified animal charcoal to convert the whole into a powder, which is
then exjiosed in a vacuous desiccator until quite dry, when it is extracted in a
Soxhlet apparatus with dry acetic ether. This solvent slowly removes the glucoside,
leaving behind nearly all dextrose and brown extractive matter. On distilling oft’
the solvent a syrupy residue is left, which if necessary is again treated in the same
manner ; usually, however, it crystallises after standing in a vacuum over sulphuric
acid for a few days. The substance may be recrystallised from hot alcohol or boiling-
water.

The glucoside crystallises from water in brilliant leaflets, and from alcohol in small,
transparent, rectangular prisms. It has no definite melting point, becoming brown
when heated much beyond 100°, decomposing completely at 200°. It is easily
soluble in hot alcohol, hot acetic ether and boiling water, separating in crystals on

ON CYANOGENESIS IN PLANTS.

403

cooling. It is however retained in solution by aqueous solutions of dextrose, a
peculiarity which accounts for the great difficulty we at first experienced in isolating
it from the plant.

It appears to contain water of crystallisation, since it loses weight when heated for
some time in a water oven, but the amount cannot be accurately determined owing
to the decomposition which occurs when the substance is heated near 100°.

Some trouble was met with in obtaining the material in a satisfactory state for
analysis owing to the difficulty of removing the water of crystallisation without
causing decomposition.

The following combustions were made : —

A. Material recrystallised from alcohol and dried until of constant weight in a
vacuous desiccator over sulphuric acid.

•0961 gramme gave '1887 gramme C02 C 53 -6 per cent.

•0572 „ ILO H 6-5

•1385 „ „ -2698 „ C02 C 53T

•0885 „ HoO H 7-07

B. Materia] recrystallised from water and dried at the ordinary atmospheric
temperature on filter paper.

•1260 gramme gave ‘2323 gramme C02 C 50 ‘2 9 per cent.

•0736 „ H20 H 6-42

C. Material recrystallised from alcohol and dried in a current of warm air at 80°
to 90°C.

'1021 gramme gave ’2051 gramme C02 C 5 4 '7 per cent.

•0452 „ HoO H 4-9

C14H1707N.C2H50H requires C 53 '7 H 6-44 per cent.

C]4H1707N.HoO „ C 5T1 H 5-8

ChH1707N „ C 54-0 H 5-5

C20H27O12N „ C 50-74 H 5-7

The glucoside therefore has the composition represented by the formula
C14H1707N, but when crystallised from alcohol or water the crystals which separate
contain one molecular proportion of these solvents.

For the glucoside thus isolated from Egyptian Dhurra we propose the name
dhurrin.

Hydrolysis of Dhurrin by Acids. Formation of Prussic Acid, Parahydroxybenz-

cddehyde and Dextrose.

When an aqueous solution of dhurrin is warmed on the water-bath with dilute
hydrochloric acid, hydrocyanic acid is almost immediately evolved. If the heating is

3 F 2

404

PROFESSOR WYNDHAM R. DUNSTAN AND DR. THOMAS A. HENRY

continued for some time, the liquid becomes considerabty discoloured owing to the
further action of the acid upon the products of hydrolysis. In addition to prussic
acid, a sugar and a substance soluble in ether are produced.

Paraliydroxybenzcddehyde.

About 2 grammes of the dhurrin were dissolved in 50 cub. centims. of distilled
water, and to the solution 10 cub. centims. of dilute hydrochloric acid were added, and
the mixture heated on the water-bath for 5 minutes. The liquid was then extracted
with ether and the ethereal solution dried and distilled. The residue was a brownish
oil which, on standing, solidified to a mass of rosettes of needles. The crystals were
dissolved in a small quantity of hot water, the solution filtered to remove resin, and
cooled, when the substance separated in almost colourless needles, which could be
picked out from a small quantity of the brown resin still adhering to them. After a
second recrystallisation the melting-point remained unchanged at 118°. The substance
is soluble in hot water, alcohol and ether. In aqueous solutions ferric chloride
produces a purple coloration, and bromine water a white precipitate, which becomes
crystalline on standing ; phenylhydrazine produces an immediate crystalline precipitate.
When heated in a dry test-tube the substance melts and sublimes in needles on the
cooler parts of the tube ; the vapour lias a pleasant aromatic odour.

A combustion of the purified material, dried at 100°, gave the following results : — -

•1267 gramme gave '3196 gramme CCA, 68 '7 per cent, carbon.

•0529 ,, IUO, 5T3 ,, hydrogen.

C7H602 requires C 68 ’8, H 4 "91.

The substance has therefore the composition of parahydroxybenzaldehyde.

Owing to the small amount of material available, the action of bromine on this
compound could only he studied by the addition ot excess ot bromine water to dilute
solutions of the substance, a method of investigation which, as the sequel shows, gave
rise to rather unexpected results. Under these conditions an amorphous precipitate
is formed which soon crystallises in colourless needles, forming after recrystallisation
from alcohol felted masses of needles melting at 92°, and having all the properties of
tribromphenol.

When a saturated aqueous solution of phenylhydrazine is added to a similar solution
of the substance, a crystalline hydrazone is immediately formed, which is insoluble in
ether and chloroform but soluble in hot alcohol. By operating in dilute solutions, a
well-crystallised product is obtained, melting at 178°. It crystallises from hot alcohol
in white needles which, on drying at 100°, become slightly green.

A combustion of the hydrazone gave the following result : —

•1117 gramme gave C03, ‘3018 gramme 73 '6 per cent, carbon.

ELO, "0613 ,, 6‘07 ,, hydrogen.

C6H4(OH)CH:N-NHC6H5 requires C 73‘52 H 5'6.

ON CYANOGENESIS IN PLANTS.

405

The ether soluble hydrolytic product of dhurrin is therefore undoubtedly para-
hydroxy benzaldehyde, which melts at 1 1 8°, gives a purple colour with ferric chloride,
and forms a colourless hydrazone melting at 178°.

The occurrence of tribromphenol amongst the products obtained by brominating
parahydroxybenzaldehyde has been observed by Werner (‘ Bull./ 46, 278), but it
does not appear to have been previously noticed that, by using bromine water and
dilute aqueous solutions of the aldehyde, the latter is converted almost exclusively
into tribromphenol. This result was confirmed with a specimen of the aldehyde
prepared from phenol and carefully purified from all traces of the latter.

C6H2(OH)CHO + HoO + 4Br2 = C0fLBr3OH + 5HBr + C02.

Dextrose.

The acid liquid, after removal of the parahydroxybenzaldehyde, was mixed with
powdered animal charcoal, warmed for several hours, and filtered. It was slightly
yellow, but was sufficiently transparent for observation in a polarimeter, when it
showed a marked dextro-rotation. It was next heated with phenylhydrazine for an
hour on the water-bath, and the separated osazone collected and recrystallised from
hot alcohol, when the characteristic bright yellow needles of glucosazone melting at
204° were obtained. The sugar produced is therefore d-glucose, that is, ordinary
dextrose.

The Hydrolysis of Dhurrin and its Chemical Constitution.

Hydrolysis of Dhurrin by Emulsin. — About 1 gramme of the glucoside was
dissolved in cold water, and a filtered extract of sweet almonds added, the mixture
being then set aside for 12 hours at the ordinary temperature. After a few minutes
the odour of hydrocyanic acid was perceptible, and at the end of the experiment
over 90 per cent, of the possible quantity of parabydroxybenzaldehyde was obtained.
This method of hydrolysing the glucoside is to be preferred to that involving the use
of acids, since the aldehyde produced is more easily purified.

The quantitative determination of the acid hydrolytic products of dhurrin
appeared to afford a method of confirming the formula assigned to this glucoside
from the results of combustion, which are perhaps not completely satisfactory owing
to the difficulty of obtaining the substance anhydrous without decomposing it.
Attempts were therefore made to determine the amounts of hydrocyanic acid and
dextrose produced on hydrolysis. For this purpose a weighed quantity of dhurrin
dissolved in water was placed in a small Jena flask, and sufficient dilute (10 per
cent.) hydrochloric acid added.

The flask was then corked and secured by wire and heated in a water-bath for
5 minutes. In this way complete hydrolysis is secured without much secondary
decomposition. The prussic acid formed was distilled off in a gentle current of steam,

406

PROFESSOR WYNDHAM R. DUNSTAN AND DR. THOMAS A. HENRY

collected in alkali and titrated. The sugar in the residue was estimated gravi-
metrically by reduction of Fehling’s solution.

T56 gramme gave a distillate requiring 2 ‘5 8 cub. centims. — silver nitrate

= 8'9 per cent. HCN.

T702 gramme cuprous oxide = '085 gramme dextrose = 5 4 ’5 per cent.
C14H1707N requires 8 '6 per cent. HCN and 57 T per cent, dextrose.
C20H27O13N „ 6-01 „ HCN and 80 T

The formula CuH1707N for dhurrin is therefore confirmed, and the hydrolysis
of dhurrin by emulsin, or by dilute acid, may be expressed by the equation

CuH1707N + H30 = C7Hg02 + C6H1206 + HCN.

Alkaline Hydrolysis of Dhurrin, Dhurrinic Acid. — When the glucoside is warmed
with aqueous alkalis, it dissolves, with the evolution ot ammonia, but no dextrose is
formed. On evaporation the solution leaves a sticky hygroscopic residue which
cannot be induced to crystallise.

When the hydrolysis is carried out in alcoholic solution by adding a solution of
sodium in absolute alcohol to a similar solution of the glucoside, a precipitate forms
after a few minutes, consisting of the sodium salt of the acid corresponding to dhurrin,
which is its nitrile. This acid may therefore be called dhurrinic acid. The sodium
salt is highly hygroscopic, it absorbs moisture and becomes gummy when removed
from the dry alcohol, and is therefore difficult to free completely from the accom¬
panying sodium carbonate. The free acid is almost more intractable than the sodium
salt, and, so far, has only been obtained as a syrup containing sodium chloride.
Recourse was therefore had to an examination of its decomposition products in order
to establish its constitution.

Hydrolysis of Dhurrinic Acid. — A quantity of the crude sodium salt, prepared as
above described, was dissolved in water and dilute hydrochloric acid added. The
mixture was heated on the water-bath for an hour, and when cold extracted several
times with ether. The ethereal solution was dried and the solvent removed by
distillation, leaving a brown oil, which after several days deposited minute trans¬
parent needles. These were dried by absorption of the viscous mother liquid in a
porous tile.

The substance thus obtained is at first colourless, but in a few days becomes
slightly brown. It is soluble in boiling water, alcohol, and ether, and after recrystal¬
lisation melts at 180°. With ferric chloride in aqueous solution it gives a slight
brown coloration, and with bromine water a precipitate, which after recrystallisation
from alcohol melts at 185°.

The acid liquid after extraction with ether strongly reduces Fehlixg’s solution,
and therefore probably contains dextrose.

The yield of the crystalline hydrolytic product furnished by the hydrolysis of

407

ON CYAN OGENESI S IN PLANTS.

dhurrinic acid is so small that sufficient material for analysis and identification could
not be obtained. A small quantity of the acid was converted into silver salt, and a
weighed quantity of the latter ignited, with the following result : —

•1058 gramme gave ‘0399 Ag = 3 8 '6 5 per cent., CsH704Ag requires 39 ‘05 per cent.

It seemed highly probable that the alkaline hydrolysis of dhurrin with the
formation of dhurrinic acid, and the decomposition of the latter by dilute acids, might
be strictly comparable with the similar reactions of amygdalin, which, when hydro¬
lysed by alkalis, furnishes amygdalic acid, this acid by heating with dilute acids
being hydrolysed into mandelic acid and dextrose.

Amygdalin.

C'20H27NO1]

Amygdalic acid and
ammonia.

c30h28o13 + nh3

Mandelic acid + 2 mols.
dextrose.

C8H803 2C6H1206.

Dhurrin.

c14h17o7n

Dhurrinic acid and
ammonia.

c14h1809 + nh3

Parahydroxymandelic acid
+ 1 mol. dextrose.

c8ha + c6h1206.

On this analogy the crystalline hydrolytic product of dhurrinic acid would be
parahydroxymandelic acid. We have established the identity of the two substances
by comparing the hydrolytic product with parahydroxymandelic acid prepared by
the hydrolysis of the cyanhydrin of parahydroxybenzaldehyde.

As parahydroxymandelic acid is now prepared for the first time, the following
outline of the process employed may be given.

Preparation of Parahydroxymandelic Acid. — Ten grammes of parahydroxy¬
benzaldehyde were dissolved in 50 cub. centims. of boiling water, and 30 grammes of
potassium cyanide added to the solution, which was then cooled in a freezing mixture
and 50 cub. centims. of strong hydrochloric acid gradually added, the whole being-
set aside for about 12 hours. The mixture was extracted with ether, the latter
being allowed to spontaneously evaporate, leaving an oily residue, which was mixed
with 20 cub. centims. of strong hydrochloric acid and sufficient alcohol to keep it in
solution. This mixture was boiled for 3 hours, neutralized with sodium carbonate,
filtered from the large quantity of resin formed, and extracted with ether in order
to remove unaltered aldehyde. The residual liquid was then made acid with dilute
sulphuric acid and extracted with ether until exhausted. The solvent was then
distilled off, the oily residue boiled with water, to which a little animal charcoal had
been added, and the filtered solution evaporated in a vacuum. After several days
rosettes of needles appeared in the oily residue, and these after recrystallisation from
alcohol melted at 180°, and further resembled the acid obtained from dhurrin in
giving a brown coloration with ferric chloride, and a crystalline bromine derivative
melting at 185°.

The yield of parahydroxymandelic acid furnished by the process described above is

408

PROFESSOR WYNDHAM R. DUNSTAN AND DR. THOMAS A. HENRY

only about 1 per cent., as this acid is readily converted by hydrochloric acid into a resin
dissolving in alcohol with a fine purple colour. The cyanhydrin of parahydroxy-
benzaldehyde is also very unstable, being easily hydrolysed by water into prussic
acid and the aldehyde, so that in each experiment about 50 per cent, of the latter is
regenerated. Attempts were made to utilise anisaldehyde cyanhydrin for the
preparation of the acid, but although this substance is somewhat more stable than its
lower homologue, a considerable loss occurs in decomposing the methoxy-mandelic acid
first formed.

A small quantity of the silver salt prepared from the acid obtained as described
above gave the following results on analysis : — -

•1286 gramme gave ’1642 gramme C02, C 3 4 ‘83 per cent.

•0308 „ HoO, H 2-67

‘1286 gramme gave a residue of silver weighing '0506 gramme = 39 ’35 per cent.

C8H704Ag requires C 34 ’9, H 2 ‘48, Ag 3 9 '09 per cent.

The properties and reactions of dhurrin, as described in the foregoing paragraphs,
may for convenience be summarised as follows : —

(1) The glucoside is hydrolysed by emulsin and dilute acids into parahydroxy-

benzaldehyde, hydrocyanic acid, and dextrose.

(2) It is decomposed hydrolytically by alkalis into dhurrinic acid and ammonia.

(3) Dhurrinic acid is hydrolysed by dilute acids into parahydroxymandelic acid

and dextrose.

These reactions we believe are fully accounted for by assigning to dhurrin the
constitution of a dextrose ether of the cyanhydrin of parahydroxybenzaldehyde,
which may be represented by the formula given below

HO

CuHnOj

CH.O.CoHnOo

H HO
CN

Dhurrin

Hydrolysis by acids and emulsin Hydrolysis by alkalis

(prussic acid, dextrose, parakydroxy- (dhurrinic acid and ammonia),
benzaldehyde).

Dhurrin is therefore the parahydroxy-derivative of the glucoside of mandelic
nitrile which was prepared by Fischer by the partial hydrolysis of amygdalin
with invertase, and resembles this glucoside in the ease with which it undergoes

ON CYANOGENESIS IN PLANTS.

409

hydrolysis by emulsin. It is the first member of the class of dextrose ethers
(glucosides) of cyanhydrins which has so far been found in nature, amygdalin and
lotusin being maltose derivatives.

The Enzyme of Sorghum vulgare.

In the introduction to this paper attention has been drawn to the fact that the
plant when moistened with cold water evolves hydrocyanic acid, whilst it no longer
does so after exposure to a temperature of 100°, nor is the acid formed when the
plant is placed in boiling water. These results point to the presence in the plant of
an enzyme, destroyed by heat, which has the power of hydrolysing dhurrin. This
enzyme was isolated by extracting the finely-ground plant with cold water, and
evaporating the extract so obtained in a vacuous desiccator over quicklime to remove
as much hydrocyanic acid as possible. The activity of this extract was then tested
by the addition of small quantities to solutions of amygdalin, salicin and dhurrin,
these experiments being controlled by the addition of boiled and filtered dhurra
extract to similar solutions of these glucosides.

In all three cases the glucoside was quickly hydrolysed, the formation of benzalde-
hyde, saligenin, and parahydroxybenzaldehyde respectively being recognized by the
usual tests for these substances. Comparative experiments in which the action of an
extract of sweet almonds was tried side by side with the dhurra enzyme on the same
glucosides, showed that the two extracts behaved in precisely the same way.
Similar preparations made by precipitating aqueous extracts of sweet almonds and
dhurra with alcohol and by precipitating calcium phosphate in such extracts, showed
no difference of activity in effecting the hydrolysis of salicin. The glucosidolytic
enzyme of Sorghum vulgare therefore performs the same functions as the enzyme
emulsin which occurs in sweet almonds, and in the present state of our knowledge
of the chemistry of enzymes, the two substances may provisionally be regarded as
identical.

The Cyanogenetic Constituents of Plants.

Besides lotusin and dhurrin, the glucosides we have isolated from young plants of
Lotus arahicus and Sorghum vulgare respectively, only one other cyanogenetic
glucoside is definitely known, that is, the amygdalin derived from bitter almonds,
which, however, is found in the seeds of the plant.

The results of our investigations have rendered it probable that the production of
prussic acid in a number of other plants may be associated with the ijresence of
cyanogenetic glucosides. Moreover, the question of the occurrence of prussic acid,
and the part played by it in vegetable metabolism, involves problems of the first
importance in vegetable physiology, with which we intend to deal when we have

VOL. CXCIX. — A. 3 G

410 PROFESSOR DUNSTAN AND DR. HENRY ON CYANOGENESIS IN PLANTS.

obtained a further insight into the nature of other cyanogenetic glucosides now under
investigation. So far as Lotus ardbicus and Sorghum vulgare are concerned, it would
appear that the existence of a cyanogenetic glucoside in the young plant up to the
period when the seeds ripen at any rate may serve as an important protection to the
plant from the attacks of animals. It appears that animals, indigenous to the
countries in which these plants are native, refuse to eat them in the earlier and
poisonous stages of growth. The part played hy the glucoside in the general
metabolism of these plants and the origin and fate of the cyanogenetic group still
remain to he ascertained. The temporary presence in a plant of a considerable
quantity of a cyanogenetic glucoside, together with an enzyme capable of decomposing
it, appears to us to be a fact which must have an important biological meaning.

As so much interest attaches to the subject from several points of view, we are
engaged in investigating the constituents of other plants which furnish prussic acid.
Among them we may mention Phaseolus lunatus (seeds), Lotus australis, Manihot
utilissima , and Linum usitatissmum, as well as a number of little known plants
derived from the Colonies which have proved to he poisonous to cattle, some of which
may contain cyanogenetic glucosides. From the chemical point of view it is
important, in the first instance, to isolate these glucosides and to ascertain their
properties, composition, and molecular structure. This work we have now accom¬
plished with the glucosides of I^otus arabicus and Sorghum vulgare, which are shown
to he radically different in chemical constitution, whilst each belongs to a type
chemically distinct from that of amygdalin, the only naturally occurring cyanogenetic
glucoside hitherto definitely known.

INDEX SUP.

Baenes, E. W. — A Memoir on Integral Functions.

Phil. Trans., A, vol. 199, 1902, pp. 411-500.

Asjmptotic Expansions — General Theory, Application to Integral Func¬
tions — Exponential Function.

Baenes, E. W. Phil. Trans., A, vol. 199, 1902, pp. 411-500.

Divergent Series — Theory of Summation.

Baenes, E. W. Phil. Trans., A, vol. 199, 1902, pp. 411-500.

Functions (Integral) — Types, Genres, Orders, &c. — Functions of Pp(z), &c. —
Repeated Integral Functions.

Baenes, E. W. Phil. Trans., A, vol. 199, 1902, pp. 411-500.

Lambert’s Function — Properties and Asymptotic Expansion.

Baenes, E. W. Phil. Trans., A, vol. 199, 1902, pp. 411-500.

Maclaurin Sum Formula — -Proof of.

Baenes, E. VV. Phil. Trans., A, vol. 199, 1902, pp. 411-500

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[ 411 ]

Vlir. A Memoir on Integral Functions.

By E. W. Barnes, M.A., Fellow of Trinity College, Cambridge.
Communicated by Professor A. Pt. Forsyth, Sc.D., LL.T)., F.R.S.

Received July 25, — Read November 21, 1901.

Inhex.

Part I. — Introduction.

Section Page

1. Known types of integral functions . 414

2. Exponential function . 414

3. Gamma functions . 415

4. Elliptic functions . 415

5. Bessel’s function . 416

6-8. Questions connected with the subject of the paper . 416

, History of the Subject.

9. Development of the theory by Weierstrass, Daguerre, Poincare, IIapamard, and

Borel . 417

10. Scope of the development made by the present memoir . 418

Classification of Integral Functions.

11. Definition of an integral function. Its formal expression. The standard reduced simple

integral function with a single simple sequence of non-repeated zeros . 419

12. Definitions of genre, order and convergence-exponent . 420

13. Functions of infinite genre . 420

14. Functions of algebraic and transcendental sequence of zeros . 420

15. Functions of zero order . 421

16. Density of the zeros of a function. Character of the zero-lines . 421

17. Product of simple functions with symmetrical sequences . 422

18. Simple repeated integral functions . 422

19. Their order . 423

20. Definition of multiple integral functions. Their classification . 423

21. Other types of integral functions — ring functions . 424

22. Impossibility of quite general laws . 425

(319.) 3 G 2 17.11.02

412

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

Part II. — The Theory of Divergent Series.

Section

23. Poincare’s arithmetic theory of asymptotic series .

24. The functional theory of divergent series with finite radius of convergence

25. Domain of existence of the integral “ sum ” .

26. The series for z ~1 log (1 -z) and (1 -z)~m .

27. The allied problem. References .

28. Series which are truly asymptotic. Character of the problem ....

29. Asymptotic series of the first order and their summation ... .

30. The “ sum ” and series are arithmetically dependent .

31. Such an asymptotic series can be differentiated .

32. Series of higher orders . . .

33. Their summation by the repeated exponential process .

34. The result has arithmetic dependence .

35. Example connected with the extended Riemann £ function .

36. Some properties of an extended hypergeometric function .

37. Its application to the summation of divergent series of finite order . .

38. Extension of the theory to series of any order .

39. Example of the previous theory .

40. Contrast between the two summation processes .

41. Application of the theory to the Maclaurin sum formula .

42.

>>

>>

series ex

CO ^ I

^ - 7.11+1
n = 0

43. The re-arrangement of asymptotic series .

44. The theory of the asymptotic expansion of an integral function near infinity .

45. Such expansions may differ in form in various regions round infinity

46. Conclusion .

Page

426

427
429

429

430

430

431

432

433

434

435

436

437

438

440

441

441

442

443

445

447

448
451
451

Part III. — The Asymptotic Expansion of Simple Integral Functions.

47. Order of progress . 452

48. The Maclaurin constants and integral limits . 452

49. Their expression, when </> (11) = nf>, in terms of Riemann’s (' function . 453

50. Example of the function — - . 453

51. The remainder expressed as a definite integral . 455

52. Asymptotic expansion of the function Pp (z) . 456

53. The nature of the expansion . 458

54. Asymptotic expansion of the general function of order less than unity . 459

55. Consideration of the integral which gives rise to the dominant term . 461

56. Case when ip (/) = P (a0 + ~J + |f + • • •) an(l P > 1 . 462

57. Functions to which § 54 is applicable . 463

53. Asymptotic expansion of a function with an algebraic secpience of zeros . 464

°° 1 + ^ !

59. The dominant term of the expansion of log 11 - — . 465

L1 +

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

413

Section

60. Application to functions of zero order

61. The function II

+ en

62.

63.

64.

65.

66.

67.

68.

69.

70.

71.

79

a = l L

Deduction of Lambert’s function from that in
Note on the indefinite integral of 8 55 . . .

61

The further theory of functions of zero order . .

Functions of finite non-integral order greater than unity. The function Qp (z) . . .

Asymptotic expansion of the general function of non-integral order greater than unity.

Analogy with the former expansions .

Application to functions with algebraic sequence of zeros .

Case when the coefficients become infinite .

Functions of finite integral order not less than unity. The function Rp (~) .

Deductions of the previous expansion as a limit of former results .

Construction of an extension of the gamma function .

Asymptotic expansion of a corresponding function with algebraic sequence of zeros . .

Conclusions . . . . .

Page

466

466

467

468
468
468

471

472
472

474

475

477

478

479

480

Part IV. — The Asymptotic Expansion of Repeated Integral Functions.

75. Introduction . 480

76. Asymptotic expansion of the general simple repeated integral function of finite order less

than unity . 481

<*>["/« Yl<r~l

77. The asymptotic expansion of II ( 1 + ) , where p > cr + 1 . 483

78. Deductions from the previous expansions . 484

79. Asymptotic expansion of the general simple repeated integral function of finite non¬

integral order greater than unity . . 485

® r / z\n<T <T 2 i/XiV"1-

80. Application to II 1 + — 1 en m = i >«\u/ . 485

81. Simple repeated functions of finite integral order . 486

82. Verifi cation of the asymptotic expansion of the G-f unction . 488

83. Asymptotic expansion of a repeated function with transcendental index . 489

84. Case of integral order. Conclusion . 490

Part Y. — Applications of the previous Asymptotic Expansions.

85. The number of zeros of a function within a circle of given large radius. The ease of the

gamma function . .

<® r

86. Case of the function II 1 + -^ . . .

a = 1 L e J

87. The formula N = 1U 1 log 4> (r), when p is not integral .

/»

88. The case when p is an integer .

89. Repeated functions . .

90. Integral functions formed by the sum or product of simple functions .

91. Borel’s extension of Picard’s Theorem .

491

492

493

493

494
494
494

414

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

Section Page

92. Resemblance between an integral function and its derivative. The extension of Rolle’s

Theorem . 495

93. Pp' (z) is of the same order as Pp (?), and an expression for its zeros can be theoretically

obtained. Verification when p = 2 . 496

94. The general theorem . 497

95. The nature of asymptotic expansions which involve a parameter . 497

96. Proof of § 94 in the case of a function of finite non-integral order greater than unity with

algebraic zeros . 49S

97. If F (?) be an integral function of finite order p with real zeros, F' (?) can at most have

only p imaginary zeros . 499

98. The case when the zeros of F (?) all lie along any line through the origin . 500

99. Possibility of further applications . 500

Part I.

Introduction.

§ 1. Since the fundamental discoveries of Weierstrass, much progress has been
made with regard to uniform transcendental functions : but the advances of modern

u

mathematics appear to have included no attempt formally to classify and investigate
the properties of natural groups of such functions.

Consider, for instance, the case of transcendental integral functions which admit
one possible essential singularity at infinity. They form the most simple class of
uniform functions of a single variable, and yet of them we know, broadly speaking,
the nature of but four types : —

(1) The exponential function, with which are associated circular and (rectangular)

hyperbolic functions ;

(2) The gamma- functions ;

(3) The elliptic functions and functions derived therefrom, such as the theta

functions and Appeal's generalisation of the Eulerian functions ;

(4) Certain functions which arise in physical problems (such as x~n J:, (%)) whose

properties have been extensively investigated for physical purposes.

There are, of course, isolated examples of other types of functions ; yet, broadly
speaking, except for algebraic polynomials, the four types just mentioned comprise
the extent of our knowledge.

§ 2. Take now an example of the first type of function.

sinh 7 r \/z ®

We may write - 7^ = IT

7 r V Z

1 +

?i = l [_

n-

, and hence we have

rr

1 i.

1 +

r- — c "■

-7

uu

so that when \z\ is very large, the approximate value of II
so long as — 7r < arg z < it.

1 +

'il¬

ls (27 t) 1 z - c~ ',

MR. E. W. BARNES ON INTEGRAE FUNCTIONS.

415

That is to say, for all points in the region of 2 — qo which are not at a finite
distance from the zeros of n~l z~l sinh tt ^/z, this function admits what we may call
the asymptotic expansion ('2n)~] z_i and a similar property is true of all functions
of the first class.

§ 3. Consider next the second class of functions.

We have as the simplest example

1

r (z)

ey~ z n

1

1 + — e

\*-±r

Since the time of Stirling it has been known that, when z is a large positive
integer,

T (z) = (2? r)*.

'=* e

-■■+ 2 Bs + 1 J

s=0 2s + i . 2s +2

approximately — the terms neglected involving exponentials of a lower order than
those retained.

In 1889 Stieltjes* proved that tins asymptotic expansion is valid for all values
of 2 in the region of z — oo , excejit those which are at a finite distance from the
zeros of T_1 (2).

By a different method it is possible to establish both Stieltjes’ result and the
analogous theorem that the double gamma function!

ur1 (2) = ei] ~^+y2\ z . n ri

OO GO

1 + a

o n + u n-

nA = 0 ni — 0

t

in which n = mlb}] + m.2o) 2, admits the asymptotic expansion

log

F, (z) *ST(o)

p2 (&)j, o>2)

8S'A (2) { logUi+w2 — 2 (to + to') 7 TL

I 2 s <2d0i -L - s f1 +11+ N ( >"Vs*+d°)
+ • 2 ^ U + 2) -h m {m + 1)zm ■

This expansion was shown to be valid for all points in the plane of the complex
variable 2 near infinity, which are not at a finite distance from the zeros of the
integral function Tp1 (2).

A similar theorem is true for multiple gamma functions.

§ 4. As regards the elliptic functions and the integral functions associated with
them which constitute the third type, there are no points in the neighbourhood of
infinity which are not at a finite distance from the zeros of the function and no
asymptotic approximations are known to exist.

* ‘ Liouville ’ (4), vol. 5, pp. 425-444.

t See a paper by the Author, ‘ Phil. Trans.,’ A, vol. 196, pp. 265-387.

416

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

§ 5. The best known example of the fourth type is Bessel’s function

~ \2/a +

-

J (z) — X - —A - .

^ ’ f=Q r (/i + i) r (/i + 7i + 1)

It is evident that z-" J„ (2) is a uniform integral function.

The investigations of Poisson, # Stokes, t Lipschitz,| and Jordan, § have finally
led to a rigorous demonstration by the latter that asymptotically, when n is real,

z nJ„ ( z ) = \J z “ 1 cos jz — (n + J) [, when Tv (2) is positive,
and z~n J„ (:) = \/~~ e'" + 1 ' ? cos jz + (71 -f- |-) q 1, when T\ (z) is negative.

The complexity of this result is reduced by the transformation — z2 = t or z — 1 \/t,

00 t]X

hich gives (z) = E ^ _ . - .. _ . - — , an integral function of t.

h ny ' „=02^+»r(/i + i)r(g + 7i + ly 6

And now we have for the asymptotic value of z~"Jn ( z ) the unique expression

, 2n + 1 1 (...)

(2n)~- t~ 4 el~ + t +

which is valid for all values of arg t between — jt and tt.

This shows at once that z“"J„ (z), qua function of t, has no imaginary roots which
are not at a finite distance from the negative part of the real axis. In point of
fact, these roots are known to be real and negative when n > — 1 ;|. Hence the
asymptotic expansion for

t~ 2

e 2 J„ (e2

x>

m=o

2 2>j- + n r(/i + i) r(g + n + i)

is valid for all points in the neighbourhood of t = 00 except those which are at
a finite distance from the zeros of the function.

§ 6. The question now forces itself upon us : — “ Do all integral functions of a
single variable z admit asymptotic approximations in the. domain of z— 00 , which are
valid for all 'points but those which are in the immediate vicinity of the zeros of the
functions ?

* Poisson, •Journal cle l’Ecole Polyt.,’ vol 19, 1823, pp. 349 d seq.

t Stokes, ‘Camb. Phil. Trans.,’ vol. 9, 1856, pp. 166 et seq.

X Lipschitz, ‘ Crelle,’ vol. 56, pp. 189 et seq.

§ Jordan, ‘Cours d’Analyse,’ 1896, vol. 3, pp. 254-274.

| When u is negative and between m and m 4- 1 in absolute value, there may be a finite number (2m)
of imaginary roots of z~nJ,t (z), but these are not associated with the essential singularity. Cf. Mac¬
donald, ‘ Proc- Lond. Math. Soc.,’ vol. 29, pp. 575-584.

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

417

The present memoir is devoted to the answer of this question ; and the question is
closely connected with other subjects of enquiry.

§ 7. Soon after Weierstrass, in 1876, published his great theorem relating to the
formation of uniform functions with assigned zeros, Laguerre remarked the funda¬
mental nature of the number of terms in the exponential function which is necessary
to form the “ prime factor.” The number was by him termed the “ genre ” of the
function ; and the questions at once arose : —

“ Is the genre of a function equal to the genre of its derivative ?”

“ Is the sum of two functions of the same or different genre a function of genre
equal to the common genre or equal to the larger genre respectively ?”

§ 8. Again, by Holle’s Theorem it is known that the real roots of any algebraic
equation, <f> (x) = 0, separate, and are separated by those of <f>' (x) = 0.

Is this true when f (.r) is an integral function ?

Closely connected with this enquiry is the further one : — “ If the roots of <f> (x) = 0
are all real, are those of f (x) — 0 real, in the case when (f) ( x ) is any integral
function ?”

Again, it is evident that the more quickly the zeros of an integral function increase,
the more quickly will the Taylor’s series for the function converge. Can any con¬
nection be discovered between the magnitude of the coefficients of the Taylor’s
series and the expression for the zeros of the function it represents ? In other words,
if we are given the general term of the Taylor’s series for an integral function, can
we approximately determine the nature of its zeros ?#

All these questions fundamentally depend on the asymptotic approximation for the
function. The nature of the latter serves to classify the nature of the integral
function.

History of the subject.

§ 9. As already remarked, WeierstrassI founded the theory of transcendental
integral functions by constructing functions with any assigned zeros. Laguerre^
invented the term “ genre ” to denote the number of terms in the exponential
associated in the prime-factor — and for functions of genre 0 and 1 proved that the
real roots of the transcendental integral function (j> (x) = 0 are separated by those of
f (x) = 0.

He also proved, as Hermite§ had previously proved for =7— , that if the roots of
<f> (x) — 0 are real, those of f (x) — 0 are real, provided <f> ( x ) is of “ genre ” 0 or 1.

* This question is not formally considered in the present memoir, as the expansions which are obtained,
although they will give closer inequalities than any hitherto published, must be still further developed
before inequality can be replaced by that asymptotic equality which alone would be a complete solution
of the problem.

t Weierstrass, “ Zur Theorie der eindeutigen analyt. Funct.,” ‘ Gesamm. Werke,’ vol. 2.

+ Laguerre, ‘ Compt. Rend.,’ vol. 94, pp. 1G0-1G3, 635-638; vol. 95, pp. 828-831 ; vol. 98, pp. 79-81.

§ Hermite, ‘ Crelle,’ vol. 90, p. 336.

YOL. CXCTX. - A.

3 II

418

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

His principal proposition is, ‘ If, as 2 tends to co
found for which the limit of

<£' (2)

<M2)

a very great value of 1 2 1 can be

tends uniformly to the value zero, then cf) (2) is of genre n!

Shortly afterwards, Poincare# gave further criteria for the genre of a function, and
made the important step of pointing out that the near connection between the genre
of the function and its behaviour near infinity lead to an approximate determination
of the magnitude of the general term of the Taylor’s series for the function.

After a succession of notes by Cesaro,! Vivanti| (who proved that the derivative
of a function is of the same genre as the function itself), and Hermite,§ the subject
remained in abeyance until Hadamard,!] in a memoir crowned by the French
Academy, gave a valuable extension of Poincare’s results.

The latter had proved that in the Taylor’s series for an integral function of
genre E, the coefficient of xm multiplied by the (E + l)th root of m ! tends to zero, as in
indefinitely increases.

Hadamard proved that, if the coefficient of xm is less than ( — V , the function is, in

\m \]

general, of genre less than X. He also showed that when the coefficient of xm is of
order , where X is not an integer, the function represented by the series is of

genre E, designating by (E + 1) the integer immediately superior to X.

Finally, Borel, 11 continuing Hadamard’s researches, introduced a more precise
notion than that of genre (f § 12), and attacked the difficult problem of functions of
infinite order whose convergence is very slow.

[Note added March 2 Oth, 1902.] In his recent text-book, “ Le9ons sur les
Fonctions Entieres,”## Borel has given a valuable precis of our present knowledge of
integral functions. And a paper by MELLiNff has recently come to my notice, which
should be carefully read by all interested in the subjects with which the present
memoir deals.

§ 10. The present contribution to this interesting theory differs from previous
investigations in that it is shown to be possible to substitute actual asymptotic
equalities for the inequalities which have been previously obtained.]];

* Poincare, ‘Bull, des Sciences Math.,’ vol. 15, pp. 136-144.

t Cesaro, ‘ Compt. Rend.,’ vol. 99, pp. 26, 27.

| Vivanti, ‘ Battaglini,’ vol. 22, pp. 243-261, and 378-380; vol. 23, pp. 96-122; vol. 26, pp. 303-314.

§ Hermite, ‘ Battaglini,’ vol. 22, pp. 191-200.

|| Hadamard, ‘Liouville’ (4), vol. 9, pp. 171-215.

U Borel, ‘ Acta Mathematical vol. 20, pp. 357-396.

** Paris, Gauthier-Villars, 1900.

ft Mellin, ‘Acta Societatis Fennicae,’ 1900, vol. 29, No. 4

[ft Mellin has obtained results of this nature.]

419

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

%

The memoir deals almost exclusively with simple integral functions of finite or zero
order (vide the definitions of the succeeding paragraphs.)

I reserve the consideration of functions of infinite order, and also the results which
I have obtained in connection with functions of double or multiple sequence. The
latter form a self-contained theory, which is a natural extension of the investigations
of the present memoir. The consideration of the asymptotic expansion of integral
functions defined by a Taylor’s series is also postponed, although certain noteworthy
extensions of Hadamard’s results can be at once deduced from the present theory. #

My thanks are due to Professor Forsyth for the kind way in which he has
supplied me with references and criticism.

The Classification of Integral Functions.

§11. An integral function we define to be a uniform transcendental function with
no poles, and a single essential singularity at infinity. [Sometimes it is convenient
to include algebraic polynomials.] An integral function is thus the same as a
holomorphic function, to use the translation of Cauchy’s name ; it is the equivalent
of the French “ fonction entieref and the German “ganze Funhtion.” Every mero-
morphic function can be expressed as the quotient of two integral functions.

The most simple integral function can be written in the form

eK(z) n

71 = 1

/

where H (z) is an integral function of 2, where the zero an depends solely upon n and
certain definite constants, and where the law of dependence of an upon n is the
same for all zeros. Such a function we call a simple integral function with a single
sequence of non-repeated zeros. The law of dependence may be broken for a finite
number of arbitrary zeros in the finite part of the plane. The existence of such zeros
is equivalent to the multiplication of the transcendental function by an arbitrary
polynomial coupled possibly with an exponential function of the type ep(z), where
p ( 2 ) is another algebraic polynomial. Such terms do not substantially alter the
character of the function.

Functions of the type cH(z), where H (2) is an integral function, belong to a class
apart. The integral function which we consider we shall suppose to be deprived of
such extraneous factor.

\ PT1 A (±\

_ ,m=l ™

* The present memoir was largely written during the summer of the. year 1898. In consequence, and
in spite of rigorous revision, results may sometimes appear to be tacitly claimed as new which have since
been published in papers to which reference is made in connection with other investigations of the
memoir.

3 H 2

420

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

The standard reduced simple integral function with a single simple sequence of
non-repeated zeros is thus

n

51 = 1

p»_1 i . , V.

_ T \ «m= i m v '

We shall call this briefly a simple integral function.

^12. The quantity p„ is the smallest integer such that the series N a~p‘ is

71=1

absolutely convergent. When the convergency of the series can be assured by
taking for pd some number p independent of n, the function is said to be of finite

gem e# p. In this case, if p is a real positive quantity such that S

71=1

p + e

converges and

71= 1

—e diverges, however small the real positive quantity e be,

the function is said to be of order t p, and p is called the convergence-exponent j of the
series — , — It is sufficient that the function ctn depends uniquely

Cl i Cl o Cl ii

upon n ; if we put an = (f> (n), the quantity <£ in) is not necessarily a uniform function :
it may be a definite value of some multiform function of n.

CO 1

§ 13. When there is no finite quantity p which will make the series N - r;

converge, the function is said to be of infinite genre and infinite order. The con¬
vergency of the series can, as Weierstrass first showed, always be obtained by
taking p = n. A theorem due to Cauchy proves this at once, since

If

A

/if 7. =

0.

It is equally sufficient to take p = log n, for then S , — r l0^ =

71= 1

— V

1

7i=i n

log | a, |

; and the

latter series is convergent, since | a„ | increases indefinitely with n.

But a smaller number still is a sufficient value for p, namely, the greatest integer

contained in — , where e is any positive quantity as small as we please. §

The great difficulty in the theory of asymptotic approximations for functions of
infinite order consists in finding the minimum value of p. I do not intend to consider
such functions in the present memoir. Functions of the type cfI{:), where Id (2) is
holomorphic, are of course integral functions of infinite order.

§ 14. It is evident that if an does not increase more quickly than some (possibly
fractional) power of n, however small, the associated integral function will be of

* Laguerre, ‘ Comptes Renclus/ vol. 91; ‘ CEuvres,’ vol. 1, pp. 167 et scq.
t Borel, ‘Acta Mathematica,’ vol. 20, p. 360.

X vox Schaper, ‘ Hadamard’sclien Func-tionen,’ p. 35 ; Borel, 1 Fonctions Entieres,’ p. 18.

•§ Borel, ‘ Acta Mathematica,’ vol. 20, p. 360.

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

42]

infinite order. On the other hand, if oq increases faster than any algebraic power of
n, however large, provided it be not actually infinite, the function is of zero order.
In other words, functions whose order is “ finite both ways,” to use De Morgan’s
phrase, have zeros which are to a first approximation algebraic.

The zeros of the function are said to be actually algebraic when they are given by

on = c0n

7 P

1 + + - +

npi nPi

when p is of course positive and rational, the c’s are constants, and px, p . . . are
in ascending order of magnitude.

It is now evident that we can form a scale of integral functions ; thus, in between
functions with the algebraic zeros

a„ = nPl and an = nPl, where p.: > p{,

will come functions with zeros like

nPl log ??, np> log n . log log n and so on.*

Such functions we call simple integral functions of finite order with a single
simple transcendental sequence of zeros; or, in brief, functions of transcendental
sequence.

Tims

n

71=1

1 +

(n log ?i)3 J

is a function of transcendental sequence of order \ and genre zero.

§ 15. Functions of zero order, which must always be of transcendental sequence,

co

can be classified in the same way. The most simple is II

n=l

1 +

cn

Then we consider functions whose zeros are obtained by multiplying cn by an
algebraic function of n. The next step is obviously to introduce intermediate
functions by means of logarithmic terms, and so on. Then we introduce functions
formed from sets of zeros of still more emphatic convergence, such as

n

n=i

The range is obviously limitless.

§ 16. It is worth noticing that the density of the zeros along the (possibly curved)
line on which they lie, decreases with the increase of the convergence of the function.
The zeros of the higher functions of zero order have therefore a density which
becomes less as we go higher. The conception of the density of a function is perhaps
the most easy way of intuitively classifying it.

* The analogy of the De Morgan and Bertrand scales of convergence is almost too obvious to need
mention.

422

MR. E. W. BARNES OX INTEGRAL FUNCTIONS.

The investigation of the character at infinity of the zero-lines of simple integral
functions belongs to the theory of functions of real variables. I do not propose to
undertake it here. It is, however, evident that such lines cannot curl round
infinity when they belong to functions of finite non-zero order with algebraic zeros :*
they approach this point in a line which becomes ultimately straight.

§ 17. A function with a finite number of simple sequences of zeros can evidently
be built up of a number of functions, each with a single sequence of zeros.

The function will thus have a finite number of lines of zeros tending to infinity.

When the zeros of a function of order p are all of the same character and form m
lines symmetrically ranged round the origin, the function will he equal to a function

of t ( — zm) of order — .

Thus a function of order ^ with the sequences

o-n = n~

a,l" —

where <n3 = 1,

is given by the product II

51=1

I - w

r

which, considered as a function of z3, is of order T.

§ 18. A function, each of whose zeros is repeated a definite number of times,
h (say), is substantially the kth power of a function with the same sequences of non-
repeated zeros.

When the nth zero of a function of simple sequence is repeated a number of times
dependent upon n, we call the function in brief a simple repeated function. We
can obviously have repeated functions with any number of sequences of zeros. We
may, as before, limit our consideration to a function with a single sequence of zeros ;
such a one may be written

FW =

n

51 = 1

1 A

5.1 = 1 mV pJ

The quantity must, in order that the repetition of the zero may not be meaning¬
less, be an integral number depending upon n ; but, if we take the principal values
of the ensuing expressions, it is evident that we may get a generalised repeated
function by regarding p„ as a general function of n.

Un

The quantity p„ must be so chosen that S -r— is convergent.

5j=l anP"

* This statement does not deny that they can curl a finite number of times in the finite part of the
plane.

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

423

We must then, in general, have \J

n = x>

1

Pa

\_ann

= 0

On 1

that is to say, ann p,~n must

increase indefinitely with n. We can no longer assign log »asa value for pn, which
is always sufficient to ensure convergence, as was the case with simple non-repeated
functions.

§ 19. It is evident that we may regard the value of p for which, e being a small
real positive quantity,

% — p+e is convergent and $ — p_e is divergent

7i=l C^ii n=\ etyi

as the order of the repeated function. When p is an integer, the order is equal to
the genre : in other cases the genre is the integer next greater than p.

If the order is not infinite, and the sequence of zeros to a first approximation
algebraic, p„ must be algebraic also.

Suppose that

Lt -1 = 1 and L t ~

nr

/ = co %

then, we shall have for the determination of pn, ppn — cr > 1, or p/t > (cr -f- l)/p.

Repeated functions of infinite order will not be considered in the present memoir.

§ 20. Hitherto we have limited ourselves to integral functions which possess a
finite number of simple sequences of zeros. But we have not thus exhausted the
category of integral functions. Instead of the typical zero being a definite function
of the single number necessary to define its position in the series to which it belongs,
it may be a function of two or more numbers and belong to a doubly or multiply
infinite sequence. In such case we say that the function is a double or multiple
integral function.

Thus the Weierstrassian cr function is a double integral function, and another
function of the same category is the double-gamma function to which reference has
been made in § 3.

The multiple integral functions always have ultimately a lacunary space* in the
region near infinity. In the case of Weierstrass’ cr function, this lacunary space
covers the whole region near infinity ; for the double-gamma function this space lies
between the negative directions of the axes of w1 and oj2.

By a well-known theorem due to Jacobi,! functions of treble or higher sequence
with periodic zeros cannot exist. This theorem may be extended, and we may prove
that there must, in functions whose sequence is greater than double, be such relations

* The zeros will, of course, only crowd together indefinitely on the equivalent Neumann sphere. The
possibility, or otherwise, of summable divergent expansions is the reason for the nomenclature.

T ‘ Ges. Werke,’ vol. 2, pp. 27-32.

424

ME. E. AY. BARNES ON INTEGRAL FUNCTIONS.

among the parameters that the region near infinity is not ultimately a lacunary
space. The parameters are, of course, the constants which enter into the expression
of the general zero in the form

«nl, „2 • • • = <f> {nU «2> • • •)•

Sucli functions have been scarcely considered in analysis. The ?tple gamma function
is the most simple example which it is possible to give. The theory requires develop¬
ment, since from quotients of multiple integral functions can be built up the general
solution of a linear difference equation.

It is to be noticed that, by the coalescence of the parameters, multiple integral
functions give rise to functions of lower sequence with repeated zeros. Thus the
function^

- z 1 -by co

G(2) = F^(2^e-*--- n

arises from the double gamma function when the parameters w1 and cj.2 each become
equal to unity.

The separation of multiple functions into functions with repeated and non-repeated
zeros and their classification would be carried out on parallel lines to the process
adopted for simple functions. As, however, detailed developments of the asymptotic
expansions of such functions are not investigated in the present memoir, I do not
intend to consider such functions further.

It has been already observed that by the substitution of zm (m integral) for z, we
derive from any simple integral function a function with m times as many sequences
of zeros. The substitution of e: for z will transform a simple function into one of
double sequence. [An example of this is given subsequently (§ 62), where Lambert’s
function is derived from one of simple sequence.] By transformations of greater
complexity wre may evidently construct functions of limitless range.

§ 21. We are still far from exhausting the category of integral functions. For
instance, wTe may have ring functions, that is to say, functions whose zeros are
situated on concentric circles : the number of zeros on the circle depending upon n.

ao

We can readily see that such a function is given by the product II

71 = 1

where y (n) is a function of n which is equal to an integer for all values of n, and
where, if y (n) = r. inversely n = x(j (r), and

L t

00 .

For, the product will converge with

n

n=k

1 -

<M«)

x(<0'

* See a paper by the author, ‘ Quart. Journ. Math.,’ vol. 31, pp. 264 et seq.

MR. E. AY. KARNES ON INTEGRAL FUNCTIONS.

425

the first h — 1 terms for which | z | > <£ (n) being omitted. Thus it converges with

If 2 "j mx or

Exp.

ill m 1<£0)I

The modulus of the term inside the bracket is less than

OO 00

S? V

m = 1 n = /.•

1 n) I _

mx f n)

$ 00

x(»)

1 -

2

X (n)

1 4> 0 )

Now 1 —

</>(”)

xOO

(n — -f- 1

< 2

n=k

co ) 1ms for its greatest value a finite positive

2 X («>

converges, which is

quantity A (say). The product then converges if 1 -

n=h I 9 \n)

ensured by the condition assigned at the outset.

The function whose existence has thus been established has y (n) zeros on a circle
of radius j (f> (n) | . If, since the assigned condition makes the order of c f> (n) greater
than that of y ( n ), the zeros will ultimately be separated by arcs of infinite length.

§ 22. A little ingenuity will enable us to construct other functions of types
innumerable, among them what Borel has called functions “ cl croisscmce irre-
guliere The survey gradually forces upon us the conclusion that we cannot
expect to find any general law as to the behaviour of all integral functions near their
essential singularity which is not a disguised truism, t MM. 1 1 a dam aril; and Borel
have given laws relating to the increase of all integral functions. It seems to me that
such laws rfiust be limited to particular classes of functions, and that such delimitation
cannot be stated too explicitly. Consequently in this memoir I have taken the most
simple functions and have endeavoured to study in detail their behaviour near the
essential singularity, for I believe that by such means the progress made will be sure,
if slow.

* Borel, [‘Fonctions Entieres,’ Note III.], gives an example of such a function in the form of a
Taylor’s series.

t Such a term I should apply to M. Borer’s law “the maximum value of a function is equal to the
inverse of its minimum value on an infinite number of circles at infinity.” For this law is an immediate
consequence of the possibility of asymptotic expansions ( see Part II. of this memoir).

| Osgood (‘ Bulletin of the American Math. Soc.,’ Nov., 1898, note, p. 80) states that the analysis used
to prove IIadamard’s most general law requires revision. And it is to be noted that Hadamard

(‘ Liouville,’ 4 ser., t. 9, p. 173) assumes that </> (m) is continuous, increasing, and such that L<£ (m) + ^
constantly increases ultimately.

O

o

VOL. CXCIX.

A.

426

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

Part II.

The Theory of Divergent Series.

§ 23. The development of the theory of divergent series is an interesting instance
of the progress of mathematical thought. The beginning was purely arithmetic : to
find some approximation to the value of n !, where n is a very large integer.* In the
result it appeared that the value of log n ! could be more and more nearly calculated by

adding on successive terms of a series proceeding by powers of The error is of the

order of magnitude of the term of the series next after the one at which we stop.
And, most important fact of all, the series is divergent.

If n ! he replaced by V (n + 1), a similar result can be obtained, which holds for all
real positive values of n.

Finally, there comes the enquiry as to what meaning, if any, can he attached to the
equality in the case in which n is any complex quantity.

Other approximations undergo the same process of development, so that it becomes
necessary to try and construct a formal theory.

What we may call the arithmetic theory has been given by Poincare,! for the
case in which all the quantities involved are real : — a restriction which the author
subsequently assumes to be unnecessary.

For the more extended case, when z is any complex quantity, we may say that the

• CC (t

divergent series a0 + „ +•••+! + ••• of which the sum of the first (n + 1)

Z Z

terms in S„, will, when \z\ is very large, be an asymptotic expansion for a function
J (2) if the expression \zn (J — S„) | tends to zero, as 2 tends to infinity.

Thus, if 2 be sufficiently large, 1 2" ( J — S„) | < e where e is very small.

The error J — S„ = e/z'1 committed in taking for the function J the first n + 1
terms of the series has a modulus which is infinitely smaller than the modulus of the
error J — S„_1 = au fi- e/zn obtained by taking only the first n terms, for | an | is in
general finite, and | e \ is very small.

In view of subsequent results, it proves necessary to define the equality of the
function and divergent series for values of 2 which lie along some definite line tending
to infinity. We do not then assume that the expansion is possible all round the
point 2 = 00 .

It will be sufficient to recapitulate the results which Poincare obtains.

We may multiply two asymptotic series together by the same rules as we should
apply to absolutely convergent series.

* Stirling, 1 Methodus Differentials ’ (1730).

t ‘Acta Mathematical 8, pp. 295-344; ‘ Mecanique Celeste,’ vol. 2, pp. 12-14.

ME. E. W. BARNES ON INTEGRAL FUNCTIONS.

427

In particular, we may raise an asymptotic series to any finite power, and it will
then represent the corresponding power of the function represented by the original
series.

The term-by-term integral of an asymptotic series is equal to the integral of the
function which it represents : in brief, we may integrate an asymptotic series.

In general, we may not differentiate an asymptotic equality.

[Nevertheless, we may differentiate most of the expansions which arise naturally
in analysis, and are not constructed artificially.]

Similarly, if an asymptotic equality involves an arbitrary parameter, we may not
in general (but we may fairly safely in practice) differentiate with respect to that
parameter.

Such are the main propositions of the arithmetic theory of asymptotic expansions.

The difficulties inherent in the theory are obvious when we attempt its application.
We have, in all cases, to investigate a superior limit to the remainder of the series
after the first (n + 1) terms have been taken ; and, to do this, we must have
command, even for the most simple cases, of analytical processes of great complexity
and power.

§ 24. We proceed then to consider these series from the function-theoretic point of
view.

That is to say, on the one hand, we attempt to oive a definition to a divergent
series which shall harmonise with the development of Weiersteass’ theory, and on
the other, we enter more deeply into the nature of the essential singularity of the
function of which the divergent series is the expansion.

Suppose -first that we have a series ci0 + aq z fi- . . . + aILzn + ... of finite radius
of convergency p, so that by Caijchy’s rule, L t '{/a,, = p~Y.

n= oo

When \ z\ is greater than p, the series is divergent and our fundamental conception
of a series as a command to add in order successive terms leads to no result.

And yet, if the function which the series represents be not one which has the circle
of radius p as a line of essential singularity, the function exists outside this circle,
and admits an analytic continuation. Thus the function exists even when the series
is divergent.

Can we not then regard the series when divergent as a command to perform certain
operations which shall yield the analytic continuation of the function? We can do
so, and in an infinite number of ways.

The most simple is, perhaps, given by an extension of a process developed by
BorelA

Let the plane of the variable x be dissected by some line going from 0 to oo to the
right of the axis of y.

* “Th^orie des series divergentes sommables,” ‘ Liouville,’ 5 ser., t. 2, pp. 103 et seq. “ M4moire snr les
series divergentes,” Ann. de l’Ecole Normale Superieure, 3 ser., t. 16, pp. 1 et seq.

3 i 2

428

[MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

This line of section will render (— x): 1 = e(£-1) l0*(_x> uniform, and we shall take
that value which is real when x is real and negative. *

Then it is known that*

G (x) = c(qCq + aYcpx + . . . + a,finxn + . .

This function will be an integral function, for

o '

L t Va„c„ = U . =U~ = 0.

n = : o n = co \/ ~~h & ) n — x: p H

Consider now the integral

G{xz)e-*(-x)°-'dx.
2 sin 7T0 J x x '

This integral is equal to

2 sin 7 t9

[ S \aILcllx”z"'] e x ( — x)e 1 dx, or N anzn.

J 71 = 0 71 = 0

That is to say, when \z \ < p, the integral represents the same function as the
original series. For all values of \z\, the integral, provided it has a meaning, repre¬
sents the analytic continuation of the series. And if, when the series is divergent, we
regard it as a command to perform the processes which lead to the integral

sin 7 t9 J

G (xz) e * ( — x)0 1 dx

we shall obtain a conception of such a divergent series which is in harmony with
Weierstrass’ theory of functions.

* See a paper by the author, ‘ Messenger of Mathematics,’ vol, 29, p. 105.

ME. E. W. BARNES ON INTEGRAL FUNCTIONS.

429

§ 25. We now enquire whether the domain of existence of the integral is coextensive
with the domain of existence of the analytic function defined by the original series.
Just as the series ceases to define the function by becoming divergent, so the integral
may cease to be an adequate expression by becoming infinite.

Consider the series 1 + 2 + z2 + . . . + zn + . . .

The “ sum ” of this series, when divergent, is represented by the integral

2 sin Ve f 6 W < - *r‘ dx’ in which G 04 = j,

and the integral is taken round some contour embracing an axis in the positive half
of the 2-plane,

Make now 6 tend to unity. Then G (xz) becomes ex~, and the integral becomes

e~* (1_j) dx, taken along some line in the positive half of the 2-plane.

J 0

Suppose now that x = peL\ z = 1 -f- rd* where 6 and <j) are both in absolute value
not greater than tt. Since the axis of the integral lies in the positive half of the

77* 77*

z-plane, — — e > d > — ( -fie, where e is a positive quantity as small as we please.

The amplitude of x (z — 1) is 6 -f- <f>, and that the integral may be finite this quantity

must be such that (z — 1) is negative. Therefore — > 6 </>>^ or — *> .

These conditions can always be satisfied by values of 6 within the assigned range,
if </> does not lie between or at the limits of the range bounded by e and — e.

We thus see that the function 1/(1 — z) is represented by the series

1 + 2 + 22 + . . . + Z» + . . .

within a circle of radius unity ; and by the integral

sin 7 t6

(xzY

i=o T(n + 6)

e x ( — x)° 1 dx

for all values of 2 except those which lie on that part of the real axis between the
points 1 and oo .

00 yll

§ 2G. Similarly the series S - or its integral equivalent when it is divergent,

will represent -- z~l log (1 — z), provided 2 does not lie on that part of the real axis
between 1 and oo . And the same is true of (l — z)~m and its equivalent series, when
m is not necessarily an integer. These statements form easy examples which the
reader can at once work out for himself.

It is interesting to notice that the lines from the singularities to infinity intervene
to give uniformity to the non-uniform functions to which divergent series may

430

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

co

“ sum.” Thus the divergent series t represents the non-uniform function

z~l log (1 — z), which becomes uniform when a cross-cut is made along the real axis
from 1 to + 00 .

§ 27. Suppose now that we have any function with singularities lying outside
a circle of radius p, within which the function is represented by the convergent series

ao “h aiz d- • • • T" an%n + • • •

We may join the singularities by straight lines to infinity, each line being the
continuation of the direction from the origin to its initial point. Then within the
simply connected area thus formed we may replace the function .by a set of integrals
of the type

_■ / ^ ( G (xz) e~x ( — a-)0-1 dx.

2 sin l6 J v ' v '

Which we can therefore regard as the “ sum ” of the divergent series within the
region in question, whenever this set of integrals has a meaning.

Although in general this will not be the case, we can nevertheless, if the function
represented by the series has only a finite number of poles outside its circle of
convergence and within a circle of finite radius cr, greater than the radius of conver¬
gence p, split up the given series into a sum of others each of which, except the last,
will be divergent, but capable of being represented by an integral of the foregoing-
type, while the last series is convergent within this circle of radius cr. The circum¬
stances under which the whole series can be represented by a definite integral over
the region of its existence I hope to discuss elsewhere. The problem is bound up
with the determination of the number and nature of the singularities of a Taylor’s
series and is, therefore, connected naturally with the researches of Darboux,#
Hadamard,! Borel,! Fabry, § Le Roy,|| Lindelof,1F and LealW* * * § **

8 28. So far we have been concerned with the summation of divergent series of
ascending powers of 2 which are convergent for sufficiently small values of |z|. We
will now define asymptotic series as those which are divergent, however small \z\
may be, and we proceed to consider their summation.

At the outset we can see that the problem is essentially different from the one

* Darboux, ‘Liouville’ (1878), 3 ser., t. 4, pp. 5-56, 377-416.

t Hadamard, ‘Liouville’ (1892), 4 ser., t. 8, pp. 101-186.

\ Borel, ‘ Comptes Rendus,’ October 5 and December 14, 1896 ; December 12, 1898; ‘ Acta Mathe¬
matical 21 ; ‘ Liouville’ (1896), 5 ser., t. 2.

§ Fabry, ‘Ann. de l’Ec. Nor. Sup.’ (1896), 3 ser., t. 13, pp. 367-399 ; ‘ Acta Mathematica ’ (1 899),
t. 22, pp. 65-87 ; ‘Liouville ’ (1898), 5 ser., t. 4, pp. 317-358.

|| Le Roy, ‘ Comptes Rendus,’ October 21, 1898, and February 20, 1899.

IT Linbelof, ‘Ac-ta Societatis Fennicse,’ 1898.

** Leau ‘Liouville’ (1899), 5 ser., t. 5, pp. 365-425.

ME. E. W. BARNES ON INTEGRAL FUNCTIONS.

431

just considered. Instead of the process of summation leading to the same result,
whatever the nature of the integral process chosen, we can obtain an infinite number
of results, each associated function leading to a different function, of which the given
series may be regarded as the asymptotic expansion. For when the divergent series
is convergent for sufficiently small values of | z | , it defines a function over that area
of convergence, and any summation process can only lead to the analytic continuation
of a definite branch of that function. But a true asymptotic series has no area of
convergence, and any meaning which we care to attach to it will harmonise with
Weierstrass’ theory of functions.

The essential nature of the difference between the two kinds of series may be
brought out in another way. A series convergent for sufficiently small values
of 1 2: | represents a function regular in the neighbourhood of the origin. But any
function which a true asymptotic series can represent will have the origin as an
essential singularity. And, therefore, not only can many functions with an essential
singularity at the origin have the same asymptotic expansion, but also the same
function may have different asymptotic expansions in different areas having the
origin as apex. It is almost impossible to imagine a vagary which an essential
singularity will not possess, and this fact we cannot, throughout the whole of the
investigation, too carefully bear in mind.

Inasmuch as any means of regarding an asymptotic series leads to a result peculiar
to that means, we must choose our process with care so as to obtain the most simple
result, and, if possible, so as to ensure that our conception of such series agrees with
the arithmetic point of view by which historically they were generated.

§ 29. Suppose, in the first place, that we have given the asymptotic series

CIq -j- CLyZ -f- cgz2 -}-... -f- ctnzn

_ n/

in which, by Cauchy’s rule, L t = 00 • And suppose further that Lf ALA1 — 0.

71 = 00 71 — co

Then the associated function

G (z) — «0c0 4~ ct\Cxz + . . . -\-ancnz“ + . . .

in which cn =

, will be an integral function.

r> + 6)

It is a natural extension, then, of our previous ideas to regard the asymptotic
series as the expansion of the integral

1

2 sin 7 t6

| G (xz) e~x ( — x)e 1 dx.

and, conversely, to regard the integral as the “ sum” of the asymptotic series.

432

MR. E. AY. BARNES ON INTEGRAL FUNCTIONS.

The point 2 = 0 will be an essential singularity of the function of which the
integral is the formal expression. For certain values of 2 near 2 = 0 the integral can
probably take values which differ infinitely for the smallest change in the value of 2.
This will happen when 2 lies on a line of zeros or poles crowding to 2 = 0. Along
such lines, or quite possibly within areas of the same nature, the asymptotic series
will cease to represent the function.

Further, there must always be such lines or areas of non-representation, for the

only functions to the essential singularities of which poles or zeros or other

^ 1
1

singularities do not crowd are of the types e p, ce .... , which cannot admit of
asymptotic expansion.

We have then the fundamental result that the integral cannot represent the
“ sum of the series right round 2 = 0. There will be certain lines or areas with
2 = 0 as extremities or vertices along which the asymptotic series cannot be “ summed
by any process which we may employ : these lines or areas will differ with the
different processes, but will never be absent altogether.

There are, of course, asymptotic series of the prescribed type which can never be

ao

“ summed ” by any process which we may employ. Such a one is N aczCn, in which

n = 0

L t (c„+1 — c,) = =c and L t ^'cin = 00 . But such series will never arise naturally in

n = co v. = 00

analysis, and we do not, therefore, need to trouble about them.

§ 30. We have now to consider whether, when a series of the prescribed type is
“ summed ” by means of the process indicated, the function which results admits the
series as an arithmetically asymptotic expansion according to Poincare’s definition.
Denote by f{z ) the integral which is the result of summing the series

ao + a\z + • • • + a,i~l -f- • • ,

_ n /

in which L t \/ a n = 00 and I J -v c “ = 0.

n = x 't\ = : o %

The associated function for the series is

Cf (2) — a0c0 + «1c12 + . . . + a„cnz’1 + . . ,

and for s„, the sum of the first n terms of the series, is

«oco + «icF + • • • + 0^*3*.

w

Hence

hi ere

f(z) - _ 1 r G„+1 (z, ■?)

z“ 2 sin 7T0 J z "

c 1 ( — x)e " J dec,

-"G„+1 (2, x) = [a,fin -f a„+lc,l+l (xz) + ...].

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

433

Now 2“" G,i+1 (2, x) is an absolutely convergent series, and | z~n GJ)+1 (zx)\ tends to
zero as n tends to infinity.

Moreover z~n Gn+1 (zx) and G (xz) are functions which, while n has any finite value,
have the same character near \xz\ — oo , for they only differ by the polynomial
a0c0 + axc^zx + . . . + (zx) .

Therefore the

integral

: a [Gri + 1 e x ( — xY~n 1 dx will represent an
2 sin 7T0 J zn v ’ r

analytic function of z whenever 0 ^ j G (xz) e * (— x)e 1 dx does so; and the

two functions will have the same character near z = 0.

Therefore, within those areas for which the second integral represents the “ sum”
of the given asymptotic series, the first integral is finite, and | z~n {f(z) — 5,,} j tends
to zero as \z\ tends to zero.

Thus the asymptotic equality satisfies Poincare’s arithmetic definition. The
reader must note very carefully that this theorem does not apply to divergent series
which have a finite radius of convergence. It is necessary that 1 2 1 should tend to

zero. No computer, for instance, could make 1 — 2 -j- 23 — 23 + . . . tend to '

§ 31. Suppose now that we differentiate the series

«0 + «x2 + • . . + ClnZ’1 + . . .

in which

L t y/an = co , and L£ — o.

?l = co n—z o %

/

We shall obtain the series

cq + 2 a2z + . . . + nanzn~l . . .

If we “ sum ” this series by the exponential process (the name which it is convenient
to give to the process employed in the preceding paragraphs) we obtain the integral

^ ~ — - Gi (xz) e~-c ( — x)e~l dx, in which
2 sm 7tu J 1 v v '

Gx (xz) — al(\ + 2 a.2c.2xz + . . . + nancn (xz)d 1 + .
We thus see that, since this series is an integral function,

Gx (xz) = G (xz).

Therefore the “ sum ” of the series ctl + 2 ct.2z + • • • + nauzn 1 + . .

3 K

IS

VOL CXCIX. - A.

434

MR E. W. BARNES ON INTEGRAL FUNCTIONS.

Now so long as the integral

2 sin 7 t9

| G (xz) e z ( — x)8 1 dec

is within the regions surrounding z — 0, for which the original series can be summed,
the differential coefficient of the function which it represents is the function repre¬
sented by the integral

i

2 sin 7 t9

( — x)e 1 dx,

for we do not transgress the rules which govern differentiation under the sign of
integration.*

Therefore, within the region for which an asymptotic equality is valid, such
equality may be differentiated.

Similarly such equality may be integrated. And the process of differentiation or
integration may be repeated any number of times.

§ 32. We have hitherto limited ourselves to the consideration of asymptotic series
of the type

o0 -f- oqz + . . . -f- cinzn

_ n'/

in which L£ \/an — co and I A MTk — o.

7? = 00 71=00 ^

The first condition is necessary that the series may have zero radius of con-
vergency, that is to say, that it may be asymptotic.

The second condition was requisite in order to ensure the applicability of the
exponential process.

V

It is convenient to call an asymptotic series for which L t — T — o an asymptotic

n = cn %

series of the first order ; one for which this limit is greater than zero, but L t

n = cc

\/ an _ n

o ^

a series of the second order, and so on.

We have given in the preceding paragraphs the theory of summation of series of
the first order. But suppose that we wish to sum one of the most simple asymptotic

z (~y

n=l

noullian function.

By Cauchy’s theorem, re-discovered by Hadamard, we know that

— — where S n(a) is Hermite’s Ber¬
ne" ’ v ’

series, that for lo

F (z + a)
T(z)^ ’

namely

* Jordan. ‘ Cours cf Analyse,’ 2nd edition, vol. 2, pp. 151-157.

ME. E. W. BARNES ON INTEGRAL FUNCTIONS.

435

for the expansion

* S„ ( a )

n = l H ■

xn

is only valid within a circle of radius 277.

We see then that the asymptotic series is such that, if we denote the coefficient

of by

Lt > 0, and U = 0.

n=00 ^ n=CO ^

The series is thus of the second order,
preceding paragraphs will be

G (n) =

And the associated function formed as in the

" ( — )n_LS„0)

t - — - - ±n«

which is not an integral function.

Our analytical machinery therefore breaks down, and we must attempt to
extend it.

Just as the original problem admitted of an infinite number of solutions, so we
may now proceed in an infinite number of ways to give an analytical meaning to
asymptotic series of the second or higher orders.

Of these two would appear to be most natural. We may either use some more
powerful associated function than we used in the exponential process, or we may
repeat the exponential process until we arrive at a finite analytical function.

§ 33. Let us consider in the first place the second of these alternatives.

If we have the asymptotic series

ao + aiz + • • > + anzn + • • • >

we have agreed to say that this series is the expression of the analytic function

— — [Gj ( xz ) e~x ( — x)d~l dx, whenever this integral has a meaning, that is,

whenever Gx (xz) is an integral function, and the integral is not infinite.

Now

G*W = r^) + r(1VS)* + --- +

= a0' + a{z + . . . + an'zn -f . . . (say),

and, if the series is not absolutely convergent over the whole jfiane, we shall be
consistent with our former generalised point of view, if we regard it as determining
an analytic function

(Go (xz) e~x (— x)B~l dx,

2 sm 7T0 J ^ v 7 v 7

whenever this integral has a meaning.

3 K 2

436

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

Now

Go (2) =

-f • • • +

yji _|_

r (0) 1 ' ' 1 r (n + 0)

= a0" + . . . fi- ci,” zn + . . . (say).

If the series G! (2) had a finite radius of convergence, or zero radius of the first order,
the function Go (2) will be an integral function, and by the process just sketched,
a definite meaning has been assigned to G1 (2) and the original series.

When, however, Gx (2) has zero radius of convergency of the second or higher
order, Go (2) will not be an integral function, but we must regard the series which it
denotes as determining an analytic function

2 sin 7 t0

|G, (xz) e -r ( — x)(J 1 clx,

whenever this integral has a meaning, that is, as a preliminary condition, whenever

°R) = i% + ...+

cin

r (n + 6)

zn fi- .

is convergent over the whole plane.

The procedure may be repeated indefinitely. If we have started with an asymptotic
series which does not ultimately give rise to a function G„ (2) whose finite radius of
convergency is a line of essential singularity, we shall ultimately get an analytic
function of which the original series is the asymptotic expansion in the vicinity of its
essential singularity 2 = 0.

§ 34. The extension which we have just indicated is in harmony with the general
theory, but we have still to determine the important point as to whether the
asymptotic equality of series and functions satisfies Poincare’s arithmetic definition.

Take for simplicity the series of the second order «0 -f a1 z fi- . . . fi- auzn fi- . . . ,
for which the associated series

u0 1 tcl

r (6) ^ r (i + 6)

+ . . . +

r 0 n + 6)

U fi¬

lms finite radius of convergency and represents the function

iG (z).

The given series gives rise to the function

G (4 = j-bs jp ^ O-1 dx.

G„ (2) — a0 fi- . . . fi- a,'Z ”,

Let

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

437

then

where

G (z) — G„ (z) = * ( [XG (xz) — jG* (xz) }e x ( - x)e 1 clx,

bill 7T (7 J

Q (G\ _ V _ -L _

1 rJo r(r + 0)-

Now 2-"-1 (XG (xz) — jG,, (xz)} is an analytic function of x of the same character as
jG (xz) : hence the natures of the two functions G(z) and z~"_1 {G (z) — Gn (z)\ near
2 = 0 are substantially the same. And therefore, in general, if G (z) tends uniformly
to a finite limit as z tends to zero in any direction, z~n~l [G (z) — „G(z)} also tends
uniformly to a finite limit as z tends to zero in the same direction. That is to say,

| z~n { G (z) — nG (2) } | tends to zero as z tends to zero, so that the divergent series is
arithmetically asymptotic for the function G (z).

It is evident that a repetition of the same argument will prove the arithmetic nature
of the asymptotic dependence of a series of any order and the function to which it
gives rise by successive applications of the exponential process. But one case of
exception must be noticed. At each step the equivalence of the asymptotic series
and the derived function fails along certain lines or within certain areas radiating
from z = 0. And, since the effect of such failure is cumulative, it may happen that
before the process is finished the equivalence has failed over the whole area around
z=0. Either the series is hopeless — an artificial monstrosity that cannot arise in
practice — or we need some other process by means of which it can be interpreted.

§ 35. As an example of the process just sketched, consider the asymptotic
expansion

S n'(a) z"+1_1

s + n — 1 ’

which, qud function of z, is an asymptotic series of the second order and wherein
s and a are any complex or real parameters.

Applying the integral process associated with the exponential function to the series,
we obtain the integral

~ | G, (zx) e~xz dx,

where

G, (u) = — r (1 — S) (- 1 + 2 tyyg)

and we have, for convenience, taken the auxiliary function to be

i (~)n r (2 — ft - s)xn, ■

n= 0

so that 0 is absorbed in s.

Now Gx (u) is a series of finite radius of convergency, and the analytic function

438

ME. E. W. BAENES ON INTEGEAL FUNCTIONS.

which we take the series to represent is the function of which the series is the
expansion within the circle of convergency. The series will therefore denote, with
our present conceptions, the function

p— an

- r(i -«)

The series with which we commenced may therefore he regarded as giving rise to
the analytic function

_ -L r (1 _ s) f (_ xzy-'e-'zdx = -~T (l- s) tV~l dt

27 r v 7 J 1 - e~x‘ v 7 2?r v 7 J 1 — e~l v 7

on making the substitution t = xz.

This function admits when \z\ is small, the arithmetically asymptotic expansion
from which we started.

When z~1 is a large real positive integer, the series and integral become fundamental
in the asymptotic definition of the extended Riemann £ function.

But there can be obtained by other processes an indefinite number of analytic
functions, each of which has an essential singularity at z — 0, near which point it
admits the given series as an arithmetically asymptotic expansion. We proceed to
indicate one alternative process by which such an analytic function can be obtained at
a single step.

§ 36. For this purpose we use certain results of the theory of the connection between
linear difference and differential equations.

Consider the function

f (ot, P\i • • • , pm > ' ' ) 1

■«+...+

+ 1 . . . a + r — 1

r!p1...ptu...p1 + r— 1 ... pm + r

i(-x)r+,

It is evidently a transcendental integral function which is a solution of the
differential equation

($ + a) H - & ~k Pi — 1) • • • (d + Pm — 1)

y = o,

d

wherein the operator $ = x — .

If y be any solution of this equation, form the function

— f y ( — £c);_1 dx,

2 sin ttz J J v 7

where the contour of the integral and the prescription for ( — x)z~l are exactly those
employed in the definition of the integral for T (z) previously employed (§ 24).

On integrating by parts, we have

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

439

*/» = ~

a relation which may also be written
(z+ l)/(z+ 1) =

2 sin 7rs

2 sin 7 rz

— x)z 1 dx,

[x d (y)] ( — x)z 1 dx.

We thus see that

z(z + l - Pi) . . . (z + 1 - pm)f(z)

= (-)

»»+i

2 sin

~ [ [> (3 + px - I) . . . (S + pm - l)y] (- x)z 1 dx,

and also

(z + 1 - a)f(z + 1) = 2^ir'z f [* + a) 2/] ( — ®)* ^

Therefore, since y is a solution of the equation

£ + Pi ~ 1) • • • C& + pm ~ 1) y = — x (5- + a) y.

we have

/(*+!) = (-)

_ V«-l Pll • • • (? + 1 pm)

(z + 1 — «)

The general solution of this difference-equation is

/(*)•

r(z)

r(a — z)

r (pd . . . r (pm)

r («) ’ r (Pl — z) . . . r (Pm —

_ a ® (u a> Pi, • • • p*),

where nr (2, a, px, . . . pm ) is a simply periodic function of 2 of period unity.
We have then established the identity

2 sin irz

| Fm (a, Pl, . . . ppi, —x)(— x)z 1 dx

r(2) ro - 2) r(Pl)...r(Pm)

r («)

r (pi - 2) • • • r (pm — z)

nr (2^ oc, pj, • • • p^*)*

When a = pj, the expression on the left-hand side, and therefore that on the right-
hand side must involve p3 . . . pm only. Thus, when « = p]5 nr (2, a, pl5 . . . pffi) involves
p2 ... pm only. It must therefore be a function of a — px, ... a. — pm. Not only so,
but it cannot involve these quantities at all ; for when a = pl5 nr will be a function of
Fi — p25 • • • Pi — pm, and yet it is independent of px ; and so on when a = p2, . . . a = pOT.
Thus nr is a function of 2, simply periodic of period unity, independent of a, pl5 . . . pOT
and m.

Let m = 1, a = pj = 1 ; then (a, pl5 p3, . . . pm, -r x) becomes e~x , and the

integral becomes

2 sin

— j e x( — x)z 1 dx — T (2).

7T2 J

440

MR, E. AY. BARNES ON INTEGRAL FUNCTIONS.

Thus rrr (2) = 1, and we have finally for all values of a. pl} . . . pm and 2 the
identity

l f T7i / W \ _ 1 7 _ m / \ h (a — z) T(Pl) ...F(pm) ^

2 sin 7T2 j F“ pk,m” P ^ ( ■''' ' ''' “ ^ ^ r («) • r (Pl - 2) ... r (Pm - *y

This identity is the direct generalisation of the identity

2 sin ir:

( e~s (— x): 1 dx = T (2),

and we may therefore expect to be able to use it to extend our former process of
“ summing” asymptotic series.

§ 37. We may, in fact, show at once that we can sum any series of convergency
zero f (2) = «0 + axz + . . . + anzn + • . . , in which L t an — (n I)4, where k is any

n = x>

finite quantity.

For this purpose we put a = p1 = . . . = pm = 1 ;

| •' i • | -| ^ [7T,HFm( X )

F™ (.T) = 1 + xrx- + . . . + 7— + . . . , and we have — - - n —

mK ' 1 ( 1 !)W1 1 1 (rl)m 15 27tJ [sm Try]™

(—x): 1 dx = \Y (2)]”

Then, with our former notation, we take the auxiliary function

X (z) = 2 cnZn, where l/cn = [T (n + 0)]“ = -

l [7TmFm( — X)

n= 0

2tt J [sin 7r^J”

( — )"m XY 1 ( ~ £c)“ dx.

\

And now f(z) is defined by the integral

in which

W f q ( _ xz\ (-*)(- 1 c(r

2t r J 1 ’ [simr0r ’

G(w) = 2 (~)"maucnu\

n — 0

We take m > k, and then G (u) will be an integral function.

For L t ancH — n,l{-k ~^e~n^ ~m) + - •• ; and therefore L t \Z ancu — 0.

n= 00 n=

* In connection with the proof of this formula, the reader may with advantage refer to : —
Mellin, ‘Acta Mathematical 8, pp. 37-80; 9, pp. 137-166 ; 15, pp. 317-384.

,, ‘Acta Societatis Fennicse,’ t. 20, pp. 1-115.

Poincare, “American Journal,” vol. 7, pp. 203-258.

Pincherle, ‘ Accad. del Lincei,’ ser. iv., t. 4, pp. 694-700.

Pochammer, ‘ Mathematische Annalen,’ Bd. 38, pp. 586-597 ; Bd. 41, pp. 197-218.

„ ‘ Crelle,’ Bd. 71, pp. 316-352.

Orr, ‘Cambridge Phil. Trans.,’ vol. 17, pp. 182-199.

ME. E. W. BARNES ON INTEGRAL FUNCTIONS.

441

We can thus sum any natural series of convergency zero whose ntV coefficient is
of the same order as a finite power of n !#

§ 38. But we can go further than this : we can construct inverse functions which
will enable us to sum any series of convergency zero.

For we have seen that

i

7r~"'!lnw( X ) / ,\<l+0-l

(sin 7 rey™ K ’

dx = [r (n + 6)Jm.

Suppose, now, that we construct the function f(x) — % —

m— 1

This function will be an integral function of x, for we have

2wFsw(- x) '

(sill ird)*'1'

which is absolutely convergent for all values of | x [ .

But, if we operate by our integral on this function, we have ^ jjf(te) (— x)n+e~l dx

= S [r (n + 6)~\hn ; and the function N [r (n -f- #)]'m is infinite if ( n + 6) be

T7l= 1

1)1 — 1

positive.

^ co Jj /7T~mF0 i ( —

If, now, we take f (x) = S . . 5"8,:c - , Vvdrere bm is so chosen as to make the

J v 7 (sm TrOym

771= 1

CO t ...

series N bm [r (n + converge for all finite values of n, it is obvious that f(x)

771= 1

will itself converge for all values of x, and so be an integral function.

We may now take for the associated function

X (2) — c0 + cLz + . . . + cazu + , where cn 1 = 2 hm [r (n + 9)Jm ;

771= I

and by suitable choice of the coefficients b we may make ca vanish to an order as
great as we please.

We can then sum the series «0 + axz + ...-[- anzn + . . . , where aH is infinite
with n to an order as high as we please. In other words, we have invented the
analytical machinery necessary to sum any (natural) asymptotic series.

§ 39. As an example, suppose that we wish to sum a series a0 + axz + . . .
+ anzn + . . . , where an is infinite like where 0 < a < 1.

* This theorem corrects a mistake in my paper, ‘ Theory of the Gamma Function,’ p. 112.
VOL. CXCIX. —A, 3 L

442

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

With our previous notation we take bm = and

1 ml

m= 1

7T2mF2m ( ~ %)
ml (sin 7 rfl)2m

2 (~xf 2 —

r=0 m=l m- !

7T

1 ! sin 7 t6

2 m

I- CO

= s (-xy

r= 0

( -7 - Y~

/j \r ! sin 7 -0/ _

so that f {%) is a transcendental integral function.

Then we have f fix) (- x)n+e~ 1 dx = 2 ^r(n + g)^'" = e[m+o? _ L
2ir J ^ v v 7 m=i m !

We take the associated function

X (z) = c0 + CjZ + . . . + + • • • where c~l = e[r(ft+0)]J — I ;

and the integral function

c4r('/i)P

G (2) = a0c0 + . . . + ancnz" + . . . 111 which L t ancn = L£ ■ [r(?t+a)p = 0.

Then the sum of the series will be represented by#

| G ( — xz) f(x) (— x)e~l dx.

§ 40. We have now, by means of the generalised exponential functions, given the
machinery by which we may expect to be able to “ sum ” a natural asymptotic series
of any order.

It may be proved just as for the fundamental exponential process that the series
and the function derived from it have asymptotic equality of the arithmetic type.

Moreover, if we regard the series as having a finite radius of convergency, on
which one or more singularities lie, which has shrunk indefinitely, we, as it were,
magnify it again by means of the function F„; (x) so as to obtain the associated
function

00

G (u) = 2 ( — )nm ancnun

71=0

whose radius of convergency is infinite.

The alternative process consists in successive magnifications by means of the
function ev.

These two jirocesses will in general lead to different results : in each case we shall
obtain functions with 2 = 0 as an essential singularity ; both functions will have the

, -1

* When we take bm = (rn !)», we have

f(x) (-x)

i

2k

[r (n + fl)]2TO .

n+e-1dx= 2 7 x

m= 1 (m!),/s

and, when n is large, the series is, by a theorem due to Stokes, infinite like exp. <j^ J- [T (?j + tf)]*}

to the first approximation, We thus sum any series for which an is of order exp. { [T (»)]''}> by taking
s greater than r.

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

443

same arithmetically asymptotic exj)ansion. But the expansion in all probability will
not be valid in the two cases along the same lines or within the same areas tending
to 2 = 0. Moreover, such a result is not surprising. The original series, except
from the point of view of the computer, had no meaning ; it did not define an
analytic function over any area of the plane of the complex variable and therefore
could not uniquely represent such a function. We have, however, now given two
processes (out of an infinite number) by which we may conceive the series to define
an analytic function, and the functions thus defined each satisfy all that the
computer can demand.

§ 41. It will, perhaps, elucidate the theory which has been developed if we give two
actual examples of its application.

We will first investigate the Maclaurin sum formula, which gives an asymptotic

m— 1

value for X <f> (n) when m is large, under certain restrictions as to the nature of the

n=l

function (f> ( n ).

In the first place it is evident that such restrictions must exist : the function must
either be uniform or be limited to a definite branch of a multiform function ; and, as
2 takes increasing integral values, (f) (2) must increase uniformly.

We will assume, therefore, that <f> (2) is an integral function, which may be
represented by a Taylor’s series, a0 + apz + . . . + arzr -j- . . .

Then, if the integral be taken along a contour embracing an axis in the positive
half of the 2 plane, we shall have, by the usual expression for the gamma function,

m m . C oo

t <f> (n) = % — Hr(l+r) e~nz(- z)~r~x dz =

71=1 71=1 — ^ J r= 0

i

e 2

1 — e mz - ar T (for)
i-«-*r=o (-yr+1 c:

Suppose now that the series X arT (1 + r) zr has a finite radius of convergence p.

r= 0

CO

Then 2 arY (1 + r) ( — 2)_r_1 will be the expansion of a function convergent

r=Q

outside a circle of radius p~l.

We can always make the bulb of the contour along which the fundamental
integral is taken expand so as to entirely include this circle of radius p~l. and the
subsequent integral will then be finite.

i f €~z co CL t\

Let now Z = 0 — — — X - — Wqd2, so that Z is a definite finite quantity

"7T J 1_ 0 t=0 \

depending on the coefficients in the expansion of 6 (2).

„-(m+i)z x ar r (1 + r) ^

Then X (f) ( n ) = Z — ~ [

n= 1 47T J

1 - <T

r— 0

(-2W1

The second integral may be written in the form

3 l 2

444

ME. E. W. BARNES ON INTEGRAL FUNCTIONS.

sK1 +•

+ (-

n !

, — ni - •£
i-=0

a,T (1 + r)

(- *)r+3

clz

if we postulate that we are reversing the process by which we “sum” an asymptotic
series.

The integral is equal to the asymptotic expansion

71 = O'

S/(l)

Jl f -na | tT(1 + r) | Sh (1) r | arV(l + r)

-VJC ,.T0 (_z)r+s-» • nto 7? ! L=o F(2 +

co oo s 7 m r »

+ 1 S a,w' + 2 -^V J S a.

■ v _ _

?’=o 1 + r

r . r — 1 . . . r + 2 —

n ^

J

fw co

<f> (on) dm + -k<f> (m) + -

n =

S'„+1 (0) d *_

dh n + 1! d m

When n is odd Sb+o (0) = 0 : the integral is therefore equal to the asymptotic
expansion

Cm co , o(0) ci?n l l

j <£ im) dm + hf> (m) + *o 2);Vd / $ M

= I *<m> dm + W (’») + ,1(2^ 71) • * <”')•

We 1 lave finally the asymptotic equality^

7)1 — l

V

71= 1

</> (n) = Z + j 0 (on) dm - y> (on) + ^w„+i (m).

This equality is valid when cf> (z) is a uniform integral function of 2 such that it it

co

be expanded in the form ci0 -f- axz -j- . . . -fi- orzr -j- . . . , the series % a,T (I + r) ~r

r=0

has a finite or infinite radius of convergency.

_ 2

We must therefore have X/ a „n ! equal to a finite or zero quantity, so that a ~

must increase as fast as or faster than n. The function (f> (z) must therefore be a

function whose “ order is greater than or equal to unity.

In the particular case when the series S arY(\-\-r)zT represents an integral

r= 0

function, we may conveniently express Z in terms of the Eiemann £ functions of
negative integral argument.

* In a subsequent paper I shall show that it is better to write this formula in the form

?/? — 1
2 <f>(a

71=0

?u») = 7. +

v S'„ (a j to)
i>=o n ! dxn

nioj

in order to exhibit its analogy to more general extensions.

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

44 5

For in this case the bulb of the contour integral which expresses Z may be taken
so small as not to include the poles of , and we shall therefore have

Z = — % arr !

-7T r=0

dz

Jr

(- z)-r-i = V art(-v) = %

a

r= 0

( = 0

2< + l

(-r1 Wl

2t + 2

This series is evidently convergent if (f> (z) is an integral function whose order is
greater than or equal to 2, a condition which is equivalent to the convergency for all
values of \z\ of the series

t cirV (1 + r) zr .

It is evident that the Maclaurin sum formula will hold good in many cases in

i I log,

which (f> (z) is not a uniform function. If it be a function like zp or z9 , or either of
these functions multiplied by an integral function of order greater than unity, the
Maclaurin formula will be valid if we suitably specify the branch of the function
considered. Instead of attempting to tabulate such cases, it is perhaps better that
we should go back to the genesis of the formulae when they actually arise. Applica¬
tions of the formulae which will be made subsequently in this memoir will usually be
to cases in which <f> ( z ) has very simple values ; and all general formulae will be
tacitly supposed subordinate to what we may call the Maclaurin restrictions.

§ 42. As a second example of the theory of asymptotic series we propose to try
and find the function of which the series

+ •

+ — + .

is the asymptotic expansion near the essential singularity x = co .
We know that, if n be a positive integer,

r (n) = f e~( t'l~l dt

J 0

where the line integral is taken along any straight line L from the origin to infinity
which lies in the half of the z plane to the right hand side of the imaginary axis.
Therefore the given expansion asymptotically represents the function

CO

where G (u) is the function which is represented by the series X un, and the integral

71 = 0

is taken alone; the straight line L.

O Cu

446

ME. E. W. BARNES ON INTEGRAL FUNCTIONS.

The given series is therefore the asymptotic expansion of the function

/(*)= (L)

X

1 -

t

X

clt.

Suppose first that the real part of x is positive ; then, putting t = xz we have

r“ exll~z)

/(*) =

J o

1 -

dz.

the integral being taken along a line along which T\ (2) is positive, that is, along
which Tv (1 — 2) is negative. [These two conditions only differ when we consider a
line practically parallel to the imaginary axis and therefore initially excluded.]

~ erxy

dy , the integral being taken along

■1 V

Putting 1 — 2 = — y, we have fix) = —
a line along which P (y) is positive ; and therefore

e~z dz

f(x) = - !

(1),

the integral being taken along a line still in the positive half of the 2 plane. Thus

e z dz

+ e P z

dz —

f(x) = “

Hence, if y be Euler’s constant,*

erz — 1

dz

, where we take e | to be very small.

f(x) = log c + y -

dz — log e + log (— x) + terms which vanish with e I .

Finally, on making j e | =0,

f{x) = y + log ( — x) + f

J 0

e2 — 1

dz.

It will be noticed that the integral (l) obtained for f (x) has a pole along the line
of integration so that it has an infinite number of values, all differing by '2ttl , which
are implicitly involved in the logarithmic term.

We see then that, when the real part of x is positive, the given series is the

« . 00 xr

asymptotic expansion of the function y -f- log ( — x) + 2 — -•

r= 1 1

Take next the case when the real part of x is negative. As in the first case, the

r00 e*(i-*)

of j

series in the asymptotic expansion
for which H (2) is negative.

Jot

dz, the integral being taken along a line

Thus it is the expansion of

-CO —Til

e xy

1 y
r e~z dz

dy along a line for which H (y) is negative
along a line for which Tv (2) is positive.

* See the author’s paper, ‘ Messenger of Mathematics,’ vol. 29, pp. 98 and 99.

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

447

On pursuing the same course as before we find that the given series is the
asymptotic expansion of the function

y + log ( *) + j y 1 dz.

But there is the important difference that now the integral which has led to this
result has no pole along the line of integration. And log ( — x), instead of being
allowed to take any one of an infinite number of values, has such a value that
log ( — a?) is real when x is real and negative, and has for complex values of x whose

7T

real part is negative an amplitude which lies between + — •

We see then that the process employed has led, when 2ft (x) is positive to an
infinite number of functions, all of which have the same asymptotic expansion ; and,
when 2ft (x) is negative, to but one such function.

Evidently when we seek an asymptotic expansion for the function*

ao yj" +1

x ) = e~x % - -

' r=1 r . r !

we may say that we get, when 2ft (x) is positive,

f(x)

1 ! n !

1 + — +... + — +•

for terms like x{y + log ( — x) + 2rmn}e * are negligible compared with the least
term of the asymptotic series ; but when 2ft (x) is negative, we get

1 ! n\

f(x) = e~xx{ log (- x) - y] + 1 + —

in which successive terms are of decreasing order of magnitude.

The zeros of the function f (x) near the essential singularity x = oo , are ultimately
along the imaginary axis.

We thus have an illustration of two important propositions : —

(l.) A uniform integral function may admit of asymptotic expansions of different
form in different areas with their vertices at its essential singularity.

(2.) These portions of the plane are separated by lines of zeros of the function.

§ 43. Inasmuch as in Parts III. and IV. of this paper we proceed to actually
obtain asymptotic expansions satisfying these laws for all the most simple types of

* I was asked to investigate this function by Mr. G. W. Walker, Fellow of Trinity College, who
desired to compute it in certain physical researches. Originally I obtained the expansion by considering

the differential equation x2 + y = x, in a way bearing great resemblance to that employed by Horn,

‘ Crelle,’ vol. 120, pp. 17 and 18.

448

MR, E. W. BARNES ON INTEGRAL FUNCTIONS.

integral functions, we now proceed to sketch the process which will be adopted, and,
in the course of our outline, to prove at once the validity of that process and the
laws which govern its results.

Suppose, in the first place, that we have the absolutely convergent expansion
F (2) = a0 + cqz + . . . + oszs + . . . in which the coefficients are functions of a
variable t, asymptotically given for large values of j 1 1 by expansions of the type

an — w; + pg + ^7 H- • • • where the quantities bn_, bni, . . . are constants and n0, n1} . . .

are numbers arranged in ascending orders of magnitude and tending to + co as a
limit, the first numbers of the series being possibly negative.

Suppose that we substitute these asymptotic values of the coefficients and

rearrange the expression for F ( z ) in powers of

t

We shall obtain, when U | is large, an asymptotic equality

F («) = ¥,

K + ^102 + • ■ • + + •

+ t-,h

K + bnZ + . • • fi- + • • •

+ • • • + NT

hs + blsz + . . . + bmszm + . . .

4- • • •

This expansion will be arithmetically asymptotic : the computer would use it to
calculate F (z) for given values of 2 and t when \t\ is large.

The series which enter as coefficients will be, in all probability, divergent ; but, as
we are looking at the whole matter from the point of view of the computer, we are
at liberty to “sum” them by the methods which have been developed in the present
part of this memoir.

If, as will be the case in the applications which we subsequently make of this
theory, these series have a finite radius of convergence, we can “sum ” them each to
a definite, possibly non-uniform, analytic function ; and we shall have an expansion

co

F (2) = S

s= 0

dependence. We shall thus have obtained a unique asymptotic expansion for the
function F (2). The case in which the series of the type

/,(*)

w

liich will satisfy Poincare’s definition of arithmetically asynqitotic

bQS + blsz + . . . + bMSz"' + . . .

have zero radius of convergence does not arise. I11 such a case we should be able to
obtain an infinite number of functions, of which these series are the asymptotic
expansions, and we should have the absurdity that the asymptotic expansion of F (2)

in ascending powers of — is not unique.

i

§ 44. A function cannot, as has already been stated, be represented by the same
asymptotic expansion for all values of 2 in the neighbourhood of 2=0 0 , unless the
function is an integral function of z~1, and the series absolutely convergent.

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

449

For unless J (2) and the series a{) + j + . . . + have such a character, it

is impossible that L t

J (2) - «0 “ "

" should, as 2 approaches infinity,

always tend uniformly to zero, whatever be the argument of 2.

But, as we proceed to show, a uniform function of finite genre,” with an essential
singularity at infinity can, in general, be represented by one or more asymptotic
expansions valid for all points near infinity except those in the immediate vicinity of
the zeros and poles of the function.

Two dilferent asymptotic expansions cannot exist within the same region, and the
regions are separated by the lines or areas of zeros or poles of the function. The
theorem is true whether the function be a quotient of repeated or non -repeared
integral functions, with zeros of simple or multiple sequence.

We need only consider the case of integral functions — the general theorem will
follow, since every function of the type just
mentioned can be represented as a quotient
of two integral functions of finite genre.

The zeros of the function must proceed
according to fixed laws, and therefore, in our
diagram of the region near infinity, they will
mass themselves infinitely close together as
we approach infinity itself. They will
therefore form certain lines (not necessarily
straight) or areas of ultimate singularity.

If the areas entirely surround 2= 00 there
will be no asymptotic expansion possible.

We thus assume that there exists an area
such as co AB, non-shaded in the figure,
within which, if the radius 00 A is sufficiently small, there are no zeros of f (z).

Suppose first, that the zeros of the function form a single simple sequence, and are
non-repeated ; then it may be written

y

F (2) = eH(2) n (l — -

n=i . an

P-1 /_tV"

&n= 1 m \««/

eH(s) <f> (2), (say),

where p is the genre ’ (independent of n), and H (2) is a holomorphic function.

Suppose that 2 lies between circles of radii ja„| and j«„+1| where n is very large,
then those terms of the product <f> (2) for which \z \ < aw+1 may be written, as in the
proof of Weierstrass’ fundamental theorem,

er,C)

where P3 (2) is a function represented by a series of positive powers of 2. For those
VOL. CXOIX. — A. 3 M

450

MR. E. \Y. BARNES OX INTEGRAL FUNCTIONS.

terms which correspond to the first n zeros, we can expand log ( 1 — in the form

log- 2

log(- O - ~ -i[~) ~ ■

We thus obtain F ( z ) = eH(:> <ji (z) — ep<:\ where, when \aH\ < 1 2; ] < | a,,+l | , P (2) is
an absolutely convergent double series of positive and negative powers of 2, together
with logarithmic terms.

Now, unless 2 be in the immediate vicinity of the zeros of the function, this
expression, considered from the point of view of divergent summable series, will be

valid for all values of |z|. For, when \z\ > an, the expression of log (1 — ~ j in the
form

L t(n

+ • • ■ +

SCL,

still exists as a divergent summable series outside the circle of radius | a„ | for all points
except those near an. Therefore the form of P (2) exists continuously as \z\ increases,
provided we do not cross the line of, or come within the immediate vicinity of, the
zeros of the function. And thus, if we treat the series entering into the expression
of P (:) as series which are summable though divergent, the expansion will he
independent of n.

.Now the expansion may be written

X (pr (to)

n

where </>,. (to) is a function of m which depends also on r. Expand X (f>,. (to)

m — 1

asymptotically in a series of successive differentials of <■/>,. (n) by the Maclaurin sum
formula, and rearrange the series.

We shall get

(A) a certain series of positive and negative powers of 2, each multiplying terms
like | </v (to) dm ; and

(B) an expansion consisting of a finite number of positive and an infinite number

of negative powers of 2, each associated with a constant arising from a
corresponding Maclaurin expansion.

The other terms depend upon n and vanish identically ; the coefficient of each
Bernoui Ilian number is zero.

When we apply the processes of divergent summation which have been previously
developed, the series which forms the group (A) of terms will reduce to a definite

MR E. W. BARNES ON INTEGRAL FUNCTIONS.

45 J

(possibly non-uniform) function x(j (z) (say). The remaining (B) terms form the
summable divergent series

co A

** z? *

s=-p *

[There will only be a finite number of positive powers of z since the genre of the
function is finite. |

We have then

A,

.. t(z) + 2 Tr-

b (z) — e s=-P -

In other words

log F (z) — xJj (z)

00 Av

admits the asymptotic expansion 2 ~ valid for all points but those in the vicinity

8= —p J

of the zeros of F (z).

§ 45. The process just sketched will become much more clear when it is applied to
various particular cases as in the following pages. The proof may, by mere verbal
alterations, be extended so as to include functions of simple sequence with repeated
zeros.

A function with a finite number of simple sequences of zeros can be expressed as a
product of functions, each with a single simple sequence. The logarithm of each of
these functions will admit an asymptotic expansion, and the sum of such expansions
will be the asymptotic expansion for the logarithm of the function. But terms of the

category i fi (2) may be of different weight in different regions, separated by bands of

/

zeros, and thus the asymptotic expansions may differ in such regions, as has previously
been seen in the case of the integral function

co ^•<•+1

e~x t - — -

r= 1 r.rl

§ 46. The general theorem which has just been given may be proved pari passu for
integral or meromorphic functions with multiple sequence. We refrain from formal
proof, as the consideration of such functions is omitted from the subsequent develop¬
ment of this paper.

Neither do I make any attempt to consider functions of infinite order, or expansions
near isolated essential singularities of uniform functions. The difficulties which arise
are all subordinate to the main necessity of limiting the type of function under con¬
sideration ; it seems doubtful whether it is possible to give any general theorem
concerning integral functions and their behaviour near infinity, which will apply to
every function which can be constructed. For exceptional classes must always be
infinite in number compared with those which can be formally defined.

3 M 2

452

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

Part III.

The Asymptotic Expansion of Simple Integral Functions.

§ 47. We now proceed to consider in detail simple integral functions. After the
discussion given in Part I., we may confine ourselves to functions with a single
sequence of zeros.

We shall find that such functions divide themselves naturally into three groups : —

(1) Functions whose order is less than unity,

(2) Functions of non-integral order greater than unity,

(3) Functions of integral order greater than unity.

In connection with each group of functions with algebraic sequence of zeros we
first consider a standard tyj3e with which all functions of the group may be
compared.

These standard functions are

P,(4= n

n= 1

1 +

, where p > 1.

Q, 0) = n

1 + .Mr e

o — ..I/p +■■■+'

pn

PIP

is an integer such that p ~b 1 > p > p.

( - zf

Pp (Z) = n

n=l

1 + -vPVe-*^ + --- +

pn

, where p is > 1 and not integral, and p

, where p is an integer > 1.

For the logarithms of each of these functions we obtain in turn the complete
asymptotic expansion near 2 = oo . We then show how all functions of the same
order with algebraic sequence of zeros yield by the same method similar asymptotic
expansions. And we indicate how it is possible to apply the same methods to wide
classes of simple functions with a transcendental sequence of zeros.

$ 48. The constants which enter into the analysis arise from the Maclaurin sum
formula (§ 41), which may for our present purpose be written

w.-l rm T> r7

S '!•■ (n) = | </>’ (n) dn - i </>• (m) + — — 4>' (m) +

(2 1 + 2)! dvvLt+1

s being any integer, positive or negative.

What we have called the Maclaurin restrictions for the function <f>(z) are always

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

453

.supposed to apply. We shall call ys the Maclaurin integral-limit for f (n). We
shall also put F* = — ( <f>* (n) dn, and call F,s the 5th Maclaurin constant for <f> (n).
When s = 0, we have the formula

to— 1 Fm

t log 4> (n) =| log 4> (n) dn — \ log <f> (m) ... + ( — Y

n= 1 -1 y0

B,+1 d) **

(2 1 + 2)! clm2t+1

log ... ,

where y0 is the Maclaurin integral-limit for log c/j (n).

^ fy°

We put log F0 = ) log <f> (n) dn, and call F0 the absolute Maclaurin constant for

<t> (n)-

When 5 is a positive integer and L t \_4>~s(n)j = 0, if is evident that y_s = — oo

n= go

and F_.? = 0.

§ 49. In the particular case when <f> ( n ) = np, p being real or complex, the
Maclaurin constants are particular cases of Riemann’s £ function.

For, for all values of s.

JL 5 / — s\

TO- 1 1 1

ns ^ ^ (1 — s)ms~1 2 ms Fi V 2t ) (s + 2 t — l)ms+3<_1

When s = 1, we have L t £ (s) .

8=1 L

m=co

We have also the special values

C(o)= -b

m

1 s

log

m

r

s!

£ (.§) = /' B^+1, when s is an even positive integer,

£ (s) = 0, when s is an even negative integer,

£ (.s) = — — ---1 , when s is a negative odd integer equal to — (2t + 1).

— t “f”

We write, when s is any quantity real or complex, £( — s) = F (s), unless s = 1,
in which case we put y = F (s).

Simple Integral Functions of Finite Order Less than Un

§ 50. Before we proceed to consider the general theory of the asymptotic
expansion of functions typified by Pp (z) = II 1 + ~p , where p > 1, we will

n=l L 70 _

454

ME, E. W. BARNES ON INTEGRAL FUNCTIONS.

consider the function F (2) = IT

-,i=i

1 + u

ir

m — 1

, which is known to be equal to

A

(27T)"1 2~*{<W - }.

We have log F (2) = E log (l f a) + - log ( 1 + ~ ) , and, if m > \z > m — 1,
we obtain on expanding the logarithms

m — 1

log F ( z ) = ( m — 1) log z — 2 % log n +

71=1

m— 1

V

-,i=l

7t~

+ (^A“ +

•s:

+

V

n=m

z (_y-i 3*

a - ... + Hat + • • -

76"

STfc2*

If now we employ the arithmetic asymptotic approximations given by the

1

n~

m- 1

Maclaurin sum formula for log \(m — 1)1}, E -y , and E n~s, we obtain, in the limit

v. rn

when Jc is infinite,

',i=i

(~)r 1 B,

log F (z) = (m - 1) log 2 - 2 (771 - £) log m - m + log ^/2 tt + E l

„ y-i r ??rj+1

+ 2 E

* ( — y-1 f m~s+1 m~s , “ ( — dr1'

+

,= 1 8# [2-S+1

4. (-rv r i

•=1 at <

dm'2’

in

Is

_ V

» ( — )r_1 B, tf2'

S=1 s |_(2s — 1) rn2s 1 2 m2s r=1 27’! dm2’

m

or

log F (2) = (in — 1) log 2 — (2 in — 1) log in -f- 2m — log 2 77

+ 2 (-r1

8= 1

mif+!

. _ * _ i + v t-y \ m~

s(2s +1)2* s(2-s — 1) m'2s JJ S=1 2 s

+ s idn* 22F£2_! *

,.=1 (2r)! 1 ^

(_y-l rfSr *

_ - - - y — m~s — E - - - 777,' r

S=1 ss* diri-T s=i s dm~r

RrAAl,„-4

where we have re-arranged the terms of our double series in accordance with § 43.
Now by the theory of summable divergent series

Til"

= - log

in-

and

4 ( - y r « a

*=1 S 1 2*

. ou_ro ;

— 2 - 0- ' -f E - - - - 771~’f = 0.

m2r-l I J-2-

*=-* dm-r

Hence we have, when ??i is large, the approximation, asymptotic with regard to in,

log F (2) = in log -A + 2,in — log (2772*)

+ 22 (-WA b

2,' ' I ' -s 2s + 2 2? T Ws - 1 2s ) M-'-' |

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

455

f & ' ( -- OTIS'S 1

or log F (z) = — log 2i rzh + 2m <{ 1 -j- ^ - - - — > •

& W & 1 ^ ,=-* (2s + 1)*J

Suppose now that \z\ is large, and that

m2 = ze~ie where 6 = arg z.

[This assumes that \z\ is a large integer, a restriction which, as will he seen later,
can easily be removed.]

Then, when \z\ is large, we have asymptotically

log F (z) = — log 2772* + zhe 2

>-■ (_y
2+2'

»=-k S + 2

The sum of the Fourier’s series inside the square bracket is, when — tt < 6 < 77,
equal to ire* .

Therefore, when \z\ is large, we have asymptotically

log F (2) = — log 27 TZh + 772".

§ 51. In the preceding investigation we have assumed that the Maclaurin sum
formula expresses asymptotically the values, when m is large, of the functions

m— 1

log (m !) and 2 n2s (s positive or negative).

n = 1

Accurately we have of course

If70 d

log-m — 1 t — (m — I) log m — lo g — - — — ,

& \ 2 / & I V 1 J q e^y — 1

m— l

2 n*3 =

«= 1

m

2j+1

irt“

2s + 1 2 t Jo e2*y — 1

dy

log ( /u -f- /.//) — log (m — iy)
( m + Ly)~ ~ ( m — li/)2s , when s is positive ;

and

S _L _ 1 1 _ 1 r dy

„ = m n2s (2-s — 1) m2*-1 2m2j " 1 J0 e2^ —

positive:

Hence, in the limit when k is infinite,

(m + iy) 2,9 — (m — vy)~

■2s

when s is

log F (2) = — log (2772") + 2m ] 1 + 2

i, (-y^

s=-k.z3(2s + 1)

( -)s-1_ ( m + wT

S Z*

+ tL^^vi{-21o«G + ‘'/)+A
- 7 l j - 2 loS - ,?/) + - 77 - }

This formula is accurate and holds whatever positive integral value m may have.

456

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

Unfortunately we may not say that the sum of an infinite series of integrals is
equal to the integral whose subject of integration is the function to which the series
of subjects of integration can be “summed.” The two integrals last written down
can then only be evaluated by reducing them to an exceeding complicated extension
of the type known as Dirichlet’s integrals. The analysis is utterly intractable.

If we make m2 = ze~w, and expand the subjects of integration in powers of 2, then
we can say that the last two terms will not contribute terms whose order of
magnitude when 2 is large is comparable with that of any positive or negative power

of 2. And, as we know, the sum of these two terms is equal to log (1 — c-2^).

The formula log F (2) = tt?} — log (2irz*) is thus asymptotic exactly as the Maclaurin

in— 1

series for m ! and X n2s, from which it is derived, are asymptotic.

n= 1

That is, for large

values of [ 2 1 , the expression z~n {logF (2) — irtfi + log27r2i] for all values of n tends
to zero as 1 2 | tends to infinity. There is, in fact, Poincare’s arithmetic asymptotic
dependence.

The preceding example will serve to show the nature of the asymptotic expansions
which we can now proceed to obtain.

§ 52. We consider first the function Pp (2) = n

n=\

, where p >

1.

We have

log Pp (:) = (»i - 1 ) log 2 - p

m — 1

log n

ni — 1

+

>1 = 1

00

. . . +

. . . +

(-r

-1 ps

n

sr

(-rv

snps

t • • .. - 00 1 /)l ^

Therefore, if we substitute the approximations for log m — 1 !, X ~p, , and X nps given

n=m H ?i = l

by the Maclaurin sum formula, we shall obtain the expansion, arithmetically
asymptotic with regard to m,

log Pp (2) = (m - 1) log 2 - p
| C-r1 j mpS+1 _

s= 1 *3'' 1 ps + 1

_j_ ^ (~rv J m~ps+1

4=1 s [ps — 1

(m - |) log m - m + log ^2^ + X - ' -w

y — ] — / . —j / _L 1 1 L

-ps 00

. , V 'B'-_ (~PS

2 r=i ps + 2r — 1 \ 2 r

m-Ps- 2.- + 1

MR, E. W. BARNES ON INTEGRAL FUNCTIONS.

457

or,

log Pp ( z ) — ( m — l) log z — P {m ~ 2) log m + Pm — P l°g \/ 27t

ps 1 *- ( — )smps , 4 (-r1

k r ( _ y-i

+ m t K

s=—k

m

_ i _L x ni _i_ v

sz* ps + l] s-_k 2 sz5 S=1 sz?

+

v (~)f 1]B>- [ - P , v/ (~)s_1 />S\ «l'

r= 1

2r- 1

- £ - L V' _ V _ L

1 2r . 2r — 1 ^ s=_, ps - 2r

ps]

+ 1 \2r/ ss* J

This expansion is arithmetically asymptotic with regard to m, and the coefficients of
various powers of ~ are ultimately to be summable divergent series.

1

Let now r = \z\ be large, and such that vp lies between m and m — 1. Then the

mp .

modulus of is a quantity which is very nearly equal to unity. We proceed to
“ sum ” the series 2' — ~ , 2' v „ - , and

£ / \ 5 1 pS

%'/ ( ) ^

$ (-)-1

mps 4' (~y ™p>

= -ks • {Ps + 1)

ZS ’ j/ljfc 2s 3s

I

r *

2?-! dm~r~ i

i — p log ™ + s'

l s=-k

Write t = log — , then the first series becomes

o r

Thus

k ( _ 1

f(t) = 2' — ~ — — es((“,0)

s=—k

S . (ps + 1 )

and

8/(0 *, (- r'p

P 3£ «=_i- ps + 1

So that

3/(0 v*, (-

/w = -p » +Ar

t}

s(t—iO)

g.? (t—iB)

If now we “ sum ” the last series we obtain

3/(0

y(£) = — p -f £ — ; and therefore

f (t) = Ae p + t — i6 — p, where A is independent of t.

When t = 0,
Hence

/(0)

k ( _ y-i

f{t) = 2' e~*ie.

J v ’ 3=_i s.ffis + 1)

t / — y-i * 0 _ y

= 2' e~sle + p 2' 1 \ e~

o N

VOL. CXC1X. — A.

458

MR. E. \Y. BARNES ON INTEGRAL FUNCTIONS.

Now, by the usual theory of Fourier’s series, p + S' - 7 erm = - qp

*t, (~>5

s=-k

-si9 _

77

provided — tt < 9 < tt.

1

s + --

p

sin

7 r

Therefore

/(0)= - +

7T

7T

(? P

Sill

so that

f(t)

iO i

Sill -

77

cp p -f t — zd — p

Hence

4, (-)*-1 ^

7T ZP

s= — k S ( pS -f 1) 3s .7 T m

r Sill —

1 mP

+ log a: - p-

The second series S' is at once seen to be equal to — \ log ~ .

s=—k * 2

And, since the term by term differential of a summable divergent series is equal

to the differential of its sum, the third series vanishes identically for all positive

integral values of r.

1

Thus, when 1 2: | p lies between m and m — 1, or possibly is equal to the latter
quantity, we have the asymptotic expansion, while — tt < arg 2 < it.

log Pp (2) = (m — 1) log z — p (m — T) log m -f- pm — p log \/ 2 n

+ - zp + m log

.7 r 1 0

Sill

p

! 1 mp , 4 (~ y 1 F (ps)
pm — ^ log — + S

S=1

or,

77 l-p]og^¥- - - <--AlFW

log Pp (:) = — zp — p log \ log 2 + 2

sin- 5=1

P

st

Thus, when 1 2 1 has any large value, and — 7r < arg 2 < tt, we have the arithmetically
asymptotic expansion

n

n= 1

^ q/p

gsinir/p "

+

| (-)* 1
S=1

F (ps)

§ 53. The approximation represents an arithmetic not a functional equality. It
does not vary with the argument of 2, and it exists everywhere in the neighbourhood
of infinity except at points on or near the line of zeros of the function. Not only
so, but at points on the line of zeros of Pp (2) which are not in the immediate vicinity
of one of its zeros, both the function and the asymptotic series have arithmetic
continuity, and therefore the equality will hold at such points. These results accord
with the general theory developed in Part II.

ME. E. W. BARNES ON INTEGRAL FUNCTIONS.

459

The series for loo’

Vp(z)*(%Trf

[

exp

fpt

7 r

can be “summed” by the methods of Part IT.

sm 7 T/p

The function just written tends to zero near its essential singularity z = co , and the
same will be true of the function which we get by any process of summation. But,
in general, the function derived from

» (-r1

■S= 1

sz

F (ps)

w

. ill not be equal to the function from which the series has been obtained.

co

Since F (ps) = £ (—ps), the series is equal to N

(-)s

-i

r ( i + sP)

s=i s Air

the integral being taken round the fundamental contour of § 24,

c lx

( - A

,p8-l

(lx,

The series is thus equal to IG (xpz) — , where Cf (z) = N T (ps) z\

The series for G (z) is divergent and of order p. The integral is interesting in that,
in place of e~x, we have used (ex — l)_1 as our auxiliary of summation.

§ 54. We now pass on to consider the most general simple integral function with a
single sequence of non-repeated zeros, whose order is any number (zero included) less
than unity.

1

The function may be written F (z) = TI

n = 1

1 +

(n)_

, where N i . , xt is absolutely
»=i \Hn)\ J

convergent. The nth zero, — cf) (n), is a definite function of n and any Unite number of
given constant quantities.

Suppose that if r = (j) (n), then inversely n = xjj (r).

Let \z\ = R and suppose that m is a large integer such that m — 1 < \)j (P) < on.

>1=1

<£ (ri)

t % ~ / Z \ i /

Then log F (2) = % log ( 1 + yqry) = S log ( 1 +

0 (n)

+ 2 log ( 1 +

<K")/

so that if we expand the logarithms in convergent series we shall get

ra — 1

log F (z) = (on — l) log 2 — S log (f> (01)

n=i

m — 1

+ 2

•a = 1

'</> ( n ) c f r ( n )

2 7?

+ ■ • • +

(-y ( n )

+ . . .

+

y

n=m

( — y_l z3

+ • • • +i-T77TT +

jt> (n) '2(f)2 (n) ‘ ' ’ ’ ' $(fs (v)

Now, by the Maclaurin sum formula, if s be positive

N _ ft (n) - | ft (n) dn - ±ft (on) + yj ft (in) ...+(-)'-

m — 1

t

«= 1

B

/+!

d2l+l

•

2t + 2ldm~‘+2

ft (m) -f . . .

where y is a constant quantity, depending on 5 and the form of r/> (01), which we have
proposed to call the Maclaurin integral limit for ft (01).

3 N 2

4(30

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

If s be negative, we have

oo rm

— A (f)' ( n ) = j <f)' (n) c In — ^(f>* (n) + ...+( — )

71 = 111 J Vs

2t + 2! dwi~t+l ^ ^ ^ ‘ -

where ys = °o , there being no term independent of m on the right-hand side.
And

7)1— 1

2 log <f> (n) = log cf) ( n ) dn - | log <f> (m) — f

Bi+1

'-!=1

yo

2t + 2 ! dm*t+l

log <f> (m) -f ■ . .

Hence, in the limit when h = oo ,

rm

log F (z) — (m — I) log z -- j log cf) ( n ) dn

Vo

+ t

8= 1

* T( _ )*

y— 1 r m ( \s ~s rm

hr I P(n)dn+{-j^ I

^ J Vi J V —

— i j — log (wi) + 2

* rv _ v-i

( W» +

S=1

dn

y-8 A

(-y

-S;r

, - .n" \

+ 2 2 n r. i - log ^ <“> +

s<F (m)_

v (->' 1 <F(»0

(=0

S=1

I (-r1^

Now, when the limiting values for Jc = oo of the summable divergent series
are taken,

2

s— 1

x-y

-i

(-)8

(bs ( m ) + ,

SZs r \ / g(ps (.ni)_

- log cf) (m)

= log ( 1 + ) - log 1 1 +

J

(f> (m)j

— log cf) (in) = — logs:.

Hence, asymptotically,

Yo

k

s=l

-1

f m cf)s (n) dn

( m z'dn ~

J S 2s

LJ v* ° *

Jy-S scf>s (n)_

= (»» - 4) log * - [» log <4 ( »)];: + j

+ t

deb (n)
n — —
yo <MA>

.1 Jy» W*(w)J ?-*.

s (-)'

m ?i0s 1 (n)

lv y.-

*

# M + j

J y— s

zrn

cf)s+1 (n)

def) (n)

= ~ i log z + [> log cf) (n)]yu + 2 [f (n) w]y,

S=1 * •*

pH™) ,7/ 7- rr<M>«) /s_1 Z^'ylrU)

+ vM0f + 2(-H *«-** + (

j 6 (vA f s=l J 6 (yt) * r

' (yo)

<4 (y>)

I <f> (y — l)

ME. E. W. BAENES ON INTEGEAL FUNCTIONS.

461

Now, where \jj ( t ) is a given form, and such that we can integrate expressions like
xfj ( t ) ts~l dt ( s positive or negative, but integral) we need only carry out this process
and “sum” the ensuing series of positive and negative powers of z to obtain the
dominant term of the asymptotic expansion of log F ( z ). If, however, xfj ( t ) is not
thus formally given, we have to face the difficulty that the lower limits of the
definite integrals are different quantities. The lower limits, however, corresponding
to negative values of s, are such as to give rise to zero terms. If, then, we consider

r<t> (»o dt

only indefinite integrals of the type I xfj (t) ts — , and take care that in any trans¬
formation of these we do not introduce arbitrary additive constants, we may take
the asymptotic expansion in the form

Cy o

log F (z) - - A log z -F J log (f) (n)

dn — N

i (-r1 r

+f

S=1 Si

4 xfr (t) clt
t

fVOO d

dn

l + W'<’ + Lw!

S 55. It is the integral

8

= L t f

i=x J

yfr ( t ) dt

1 + 2'(—

s=- A * .

which gives rise to the dominant term of the asymptotic expansion of log F ( z ).

. . . .

This integral is evidently equal to L t

£•=*> J

yfr (t) dt

"(vr-fd-r

J t

i+i

Suppose now that z = reie and take i/x = log ( — j, the logarithm, when t = r,
having a cross-cut along the negative half of the real axis, so that

px = log +7 tl — l 0, where log - is arithmetic.

Then

I

= Lt i f

k= k> J

i log \ + v - e . . u sin (k + J ) ll 7

1 1 y— djji.

sm i /u.

Now the form of the dominant term I does not depend on the quantity log
which vanishes when r is sufficiently large. We have then

xf r (rtl)

T T [n~9 . r r w sill (k + A) Li ,

I = Ln lib {zet('x_,7)} . v -a--- da

t-=J 1 J smi/x .

an integral of the type first considered by Dirich let. #

* v. ‘Crelle,’ vol. 4, pp. 157, et seq.

462

ME. F, W. BARNES OX INTEGRAL FUNCTIONS.

The theory previously developed in Part II. tells us that this integral must, qua
function of z, be independent of 6 ; in other words, when — tt < 6 < n, that

L t

k = : o

( of/ {z eL

' n-e

(fi-TT)

}

sin (k + f) fju
sin h p

dp — 0.

But this is precisely Dirich let’s result : we thus have a valuable verification of
our theory.

Finally, then, the dominant term of the asymptotic expansion of log F (z) is the
function

f(z) - L t f ixfj{z&

l=X, J

(^ )} — dp.

sin \ p

Since we may evidently change the sign of i without altering the value of f (z) we
have

f(,\ _ ij - jr {se~‘ (*-/*>} sin (k + }) p ^

_ j 2 l sin b u ^ '

Now \Jj is the function inverse to <£. If, then, we suppose that £ and rj are
determined from the relation ze‘(,r_M) — (f> (£ -f- ip), principal values of inverse
expressions being taken, and £ and p being functions of z and p, we shall have finally

sin (k + \) p ,
sin h /x

There is no doubt that it is possible to construct functions cf) (n) for which the
preceding analysis will not hold good. # It would ajDpear, however, to be applicable
to most of the types of functions which would ordinarily arise, and a more accurate
investigation will need the exquisite finesse of certain developments of the theory of
functions of a real variable.

Note that, for the case in which ze‘(,r'M> = (£ — i p)p, we have established that

7 T L

— — zp .

. 7T

sin —

P

/x, as the simplest form in which we may write f (z).

f(z) = Lt

/:= oo

(}) (n)_

§ 56. The dominant term f(z) of the asymptotic expansion of log n 1 +

n= 1

takes a very simple form for the case in which \Jj (t) can, when t is large, be expanded
in descending powers of t in the form

xjj(t) = tP

+ 7 + f +

w

/here p > 1.

We have the asymptotic expansion

m — l

* We have assumed, for instance, that we can apply the Maclaurin sum formula to - ^ (m), and,

71 = 1

therefore, that the conditions of § 41 are satisfied.

MR, E. W. BARNES ON INTEGRAL FUNCTIONS.

4(13

fvn * ( _ y-i ry«

— i l°g 2 + j log (p (n) dn — S — - — j ft (n) drt

+r+ffto+M^+

1 7(— )*<* , (-yg*

2s 1

Since <£(»i ) is a very large quantity we may utilise the expansion of if* (t).
integral last written is, therefore, equal to

The

[4> («‘)

2 a,

)•= 0 j ./■

*(w) clt J 1 , i /(-)VS (-y*

1 ' ii \ Z? ^ ts

t p

— N

i

0 <£' p (;/t) r

, 4/ (-)' . ^

1 "r “ , r' 1

- s — r + -

P

—

a.

!'

7T

?"r

7T - — >•

= 2 (-)'tc -

■'■=° ! sin —

L p
> (

i

sin — L ( — z)p J
P

IT

IT

ZP

" f y°

If, then, we introduce the Maclaurin constants, log F0 = log <£ (r^) cZn,
— F, = | <f)s (n) dn, we shall obtain the asymptotic expansion

n

;s=i

i +

z

= F0z * exp <

ird

is ( — z )

</>(»_

. 7 r

sin -

. ( - *)1/p.

L P

| ( — F"1 F,

s = l SZ*

Such values of the many-valued functions introduced are to be taken as would be
indicated by the analysis.

§ 57. It is evident that the investigation of § 54 applies to all simple integral
functions whose primary factors need no exponential to ensure convergency. Thus

it includes all simple functions of order - , where p is real positive and > 1 with

algebraic zeros. It includes all simple functions with noil-algebraic zeros of the type
given by alt = [ anp -f bnPl + . . .] (lTn)a, where r and p are both real positive and
> 1, cr is positive or negative ; where lT (n) denotes log {log { . . . n] } . . . }, these
being r repetitions of the logarithm, and where p, pl5 . . are decreasing quantities

tending to — cc as a limit.

4(54

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

But, because of the validity of the Maclaurin sum formula, it includes simple
functions with very rapid convergence — such as those for which

€n c*i '

a„ — e? X function of n of lower order than ee ■ .

§ 58. We can now extend the result which was obtained in § 52, and find an
asymptotic approximation for a simple integral function with an algebraic sequence
of zeros, that is to say, of a function of which the rdh zero, — an, admits, when n is
large, an expansion of the form an — np -f- b-lnp~e 1 b.2np~eo- + . . ., where p is greater
than unity and the quantities el5 e2, . . . are real positive and in ascending order of
magnitude.

We take an — (f> (n) - r so that, by reversion of series,

I i Lm p + i _ 2ea h3 h ^ , , / x , N

n — vp - -vp + - - - - r p - -tp -f . . . = i If (r) (sav).

p pz 2 p \ / \ »/ /

Since, when s is positive, we may expand ans directly by the binomial theorem,
we have, when m is large,

ni— 1

Hi — 1

2 an = 2

,i=i ,i=i

nps -j- 61 + ^ nps 2e' -j- sbznp 5 e- .

fm

4>' ( n ) cin + F ( ps ) -j- sbl F ( ps — e2) +

AO) , |

2 ^ £ 0

W 1 V F (ps-tej+sb, F (ps-*,) + ...

(-)' B,+, d‘>»

2t + 2 ! drn?t+1

<t>' (™)

}> («) dn + F (ps) + Z (p, s ; £ l ■■■)- * + y(;

i-y b,+1 d2t+i

2t + 2 ! dm~t+l

<f)s (m),

where Z ip, s ;

e2 • • •
O &2 • ■ •

is a definite finite quantity vanishing with the quantities b,

And

>i=i

! . h h2 , h .

p log n -j- — yy -(- ■ -(- . .

which can be expressed in terms of a series of Riemann £ functions.

Again, when s is positive,

v __L_ — |‘m is/ \ i W O) s (— yB<+1 d2<+1

n=m<f>s(ri)~ \ $ W W 2 E 2f+~2l dv^t

m— 1 m — 1

2 log cf) (n) = 2

=i >i=i

= ["log 4, (n) dn + A log 2ir + 6, F (— E,) — y F ( — 2e,) + 6S F ( — %)

log O) , £ (-)*B/+1 d3m

- 2 +,?0(yT2i! los * W >

and wo shall put Z (o ; "'[ = F (-«,)- A F (_ 2£]) + /,. F (- t,) + . .

ME. E. W. BAENES ON INTEGEAL FUNCTIONS.

465

If now we substitute in the general formula

yo .

log F (z) = - i log 2 + | log <t> {n) dn - t f? f (n) dn

1 8=1

yjr (t) dt

i - s (—)*(— +

s=l

ts

we shall obtain log F (z) = — \ log z - p- log 2tt — Z ( 0 ; e_~’ ' ' '

^ \ ^1> • • •

+ 5

S=1

+ f

5=1
2 1— 2e

1-e,

'+ I &■. U- a p — ei &,

tp — — t p 4- H „ 1 ~ t p — —t p . . .

b.

dt

1 +

b (- os

And when we sum the Fourier series which result from the last integral, we find
log F (2) = — i log 2 - -J- log 2tt - Z ^0 ; ‘ *

CO ( — V 1 CO ( V 1 /

+ - F (p6‘) + 2 — — Z(p, S ,

s=l SZT s=l S2" \ 01; On, . • .

, . er e2- • •

+

7T

1 5,7r

2P -

2 p

Sill

7 r

P -L Cl

r Sill 7T - ±

TT^ . P + 1 ~ 2U

1 - * ^ 2

l-2t,

ZP

sin 7 r

1-2^

b.,7T

P

1-6

2 P.

Sill 7T

1 — 6„

§ 59. This expansion is valid for all values of any z which lie between — it and 7r.
It is arithmetically asymptotic in the same way as the expansion from which it is
derived.

We see from the results just obtained that the asymptotic approximation for

log ri

n = 1

1 + -

au

, where an — u p + b1np“ej + . . . exceeds that for log II

n= 1

1 +

nP

by

a quantity whose first term is — Z (0 ; y2’ ’ ’ when el > 1, and by a quantity

\ b l! bn, ■ • ./

whose first term is

b{7T

P

1-6
Z P

, when ex < 1.

Sill 7T

When ex = 1, the difference of the two asymptotic approximations commences with

z9

sin 7 t9

the indeterminate form

bpr

P U

bx f^(m) dt

V J 7

6=0

, which arises from the integral

1 4

(-03

s=—k

VOL. OXC1X. — A.

o O

466

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

But this integral is equal to
(—)*<£* O)

log <j> (m) +

*

N'

s=-l-

= log M - log

1 +

0 ('»<)"

+ log

1 +

<MW)_

= log

Thus, where e2 = 1, the difference of the two asymptotic approximations commences
with the term — ?qp_1 log 2, a result which may be obtained without much difficulty
by elementary algebra.

Note that — 6}p_1 log s is the expression obtained when we reject the infinite part
— bir ze

of the function — - — . — fl for 0=0, when expanded in powers of 0.
p sin 7r 1 1

■ Note also that the constant Z ( 0 ; f v ‘ ’ ’ which, when e, > 1 is the first term of

V K »■>•••/

the asymptotic expansion of the logarithm of the ratio of our two products is

cc q r.

equal to % log -p .

71= 1

By means of the formula x > log (1 + x) > ^ ^ , where x is a real quantity lying

between ffi 1, we may prove that this series is absolutely convergent when p > 1.

Application to Functions of Zero Order.

§ 60. Hitherto no example has been given of a function of zero order, although the
general investigation of § 36 applies equally to functions of this nature. In such
cases it becomes necessary to introduce Maclaurin constants of a complexity which
seems, except in special cases, beyond the reach of present analytical processes. They
can no longer, as for functions of finite order, be expressed in terms of IIiem Ann’s
£ function nor, I believe, in terms of any functions which have so far been introduced
into analysis. An example will now be given of a very rapidly converging integral
function. It obviously would serve as the starting point of a series of interesting
researches dealing with the classification of simple integral functions of zero order.

61. We propose to obtain the asymptotic expansion of the function IT

)i=i

In the notation of the general theory we have now 6 (u) = e‘\

Therefore log (n) = n ; % log (fi (n) = — — — .

By the Maclaurin sum formula, if 5 be positive,

m— 1 rm pus p>

N. e'ls = j enscln + Q, — — + ^ scms — . . .
where Cs is the Maclaurin constant corresponding to e“s, which may be determined as

cm

in § 84. If we put e"\ln = em)s, we have C, = (1 — c j_1.

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

407

If then we carry out the general process, we shall obtain the asymptotic expansion

log II

51 = 1

1 + s

— (m — 1) log z

vi

-o)+ S

S — 1

(-r1

sz“

i (-)

8-1

c.

+ S'

8= —Jc

‘rill gins'

I e ”s dn - —

, m , 4 (-V :CS , 47 (-)s 1 (ems c

= (m - 1) log z - — + - + 2 — — — + X — — - ' - -

Z L S=1 SZ s= -l- S*

As before, we have to “sum” the final divergent series. We take \z\ to be a

large quantity such that

is very nearly equal to unity, and then we consider the

X- /' Ns— 1 f giOs piOs

Fourier series S' - \ - —

«=-* sl« 2

But S' = 2 S (—y \C°S - = - i (> - w )•

s= —A- S- 3=1 S

A ( _ V t>‘es

And s' - = - ye.

s=— A 2-S ^

Therefore we have the asymptotic expansion

loff IT

71=1

1 + A

/ im i m , 4 (-)s_lc, , ,

= (w - 1 ) log z - - + + s — yy— + 1

S=1

or finally* log n

/i = i

z

1 ~h

e“

= i (log ZY - 4 log 2 + ~ + s

log7
(-r'c.

O 0

** 7T

+ 7- Gog

3=1

62. It should be noticed that if, in the function whose asymptotic expansion has

thus been obtained, we substitute e" for 2, we shall obtain the function 14

51=1

This is an integral function whose zeros are of the form

1 + «»

z = n + (2m — 1)

7TI

n = 1, 2, 3, ... 00 .

7)1 — — OC , . . . , — 1, 0, 1, ... 00

It is substantially what I propose to call Lambert’s function. The function has
properties which are a sort of mean between those of the elliptic and double gamma
functions.

We can express Lambert’s function as a product of two double gamma functions.
It is closely connected with the well-known Lambert’s series, and in terms of it we
can express in a very elegant form the coefficients of capacity of two spheres.

* The dominant terms of this result are equivalent to those given by M ELLIN, ‘ Acta Soeietatis
Fennicse,’ t. 24, p. 50.

3 O 2

468

ME. E. W. BARNES ON INTEGRAL FUNCTIONS.

§ 63. The reader will notice that in the preceding analysis we have used the
methods and not the result of the general formula.

The reason is that with an exponential subject of integration we are unable to
ensure that we do not introduce arbitrary additive constants when the indefinite
integrals are transformed as formerly.

For in this case <£ ( n ) = en and i fj ( n ) = log n ;

and we have to consider a series of integrals of which the first is

<f> (m)

p ( t ) dt

We are tempted to say that this integral is equal to

1 log [f (m) z1] dt

log (f) ( m ) + lo.

O’ t

t

whereas we only avoid introducing an additive constant by saying that

f {r3Ji = i [log0- (4 (m) t)T = J [log 4 (m)]l

§ 64. The integral function just considered is the most simple function of zero
order. In carrying out the algebraical analysis of a theory of such functions, it would
be necessary to consider the types

oo

n

n = l

n

n = 1

l

&c.

The asymptotic expansions for these successive functions are of successively lower
orders of greatness — they are never, however, of so low an order as zn, where n is
finite. This agrees with the known theorem that an algebraical polynomial is the
only uniforn function of such an order. Unfortunately, unless we introduce new
analytical functions defined by definite integrals, we cannot investigate formally
asymptotic approximations for such types ; and until the properties of such new
functions are investigated, we but express one unknown form in terms of another.

Simple Integral Functions of Finite Non-integral Order Greater than Unity.

§ 65. In the investigations to which we now proceed of simple integral functions of
finite non-integral order greater than unity, the theoretical considerations which
have been given in detail for functions of order less than unity will for the most part
be suppressed, and for brevity only the bare analysis will be written down.

We consider first the standard function Qp ( z ) = II ( 1 + Uyj e »

where p > 1 and p is an integer such that p + 1 > p > p-
Let 2 = He'9, and suppose that E. is very large.

(Nf

nil -PlP

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

4G9

Take m an integer such that m — 1 < Ilp m.

Then

m — 1

Q ,(*) = n (l +

?i=l I

n

\ « (-)V

~ Tr + . . . + -- •

.1 /p/e n'lP pnPlP

x n i +

,i/p

(-)

VP

e n1/p

so that

Qp (*)

m— 1

n

p=i

n 1

m — 1

1 + A

x n

n = 1

e ii

i-+ +<=^

i/p • • • pnviP

x n

« = 111

(-)pzp+1 (-)P+1zP+2

P+1 + P+2 +

, (/) + ])-,! P (p+2) il P

ancl hence

^ 9/1 — 1

log Qp (?) = (m — 1) log z — 2 log n

P n = 1

HI — 1

+ 2

71=1

'/l'P ( — )* 1il*'P ]

— — ~r • • ■ “r

sz"

m— 1

+ S

41 = 1

_ , l p H“ ■ • • 4"

(~W

pn

?>.p

+ l (_)/>+! 2P+2

+ +

| p+i i p+

n=lll \_{p + 1)#' 0» + 2)»p

Now, when m is a very large integer,

m~ 1 1 m *,p

v _ _ _ _ _

7>.=i /"s/p 1_«_ 2

+ . . ■ +

(-y

— -2t-

v2 1 + 2

-R/-k -fF C — -

+ i \ p

And, when s is positive and greater than p,

n=7 » nslp

wi1_,,p

1-

m~*p

•+...+

(-y

— s

n + 1

-s — 1 V P Jm2‘-p+1

p \2/ + 2/

+

We use these Maclaurin approximations and rearrange the double series which
results as the arithmetically asymptotic approximation for log Qp ( z ). We obtain,
in the limit, when the limits of the summable divergent series are taken for h infinite,
the asymptotic expansion

470

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

log Q0 («) = (m - 1) log z— 1 (m - 1) log to - — - -1 log 2tt

P P ~P

+ I' F g'

l p

s=-p

+

mp3'J p \i (-) ~ UA 1 (-)

, £ B/+i f - (2t) ! i, (— )3_1

^ Zo 2t + 2\ 1 P S

\z]

'cl2l+l p"

d,2'^

r=\

\Z\ . .

We suppose now that ^-l p is a quantity which, when m is very large, is ultimately

equal to unity. Then we may “ sum,” as before (§ 52), the Fourier series, which are
the various coefficients in the preceding expansion.

ttt *. (— )® 1 fmllp\s 7 t z° . m1 p 1 . , .

We have p 2 - — - — = 4- Log - — — , provided p is not integral,

r ■ o - -L o '■ 2 / Sill 7 Tp m * z p 1 r ’

*=— t s. p + s \

^ f — V 1 ! n\}\ \s

and provided — 7r < 0 < tt. And — tt 2 f — — j = — ^ log

Also, exactly as before, the coefficient of B^ + 1 in the asymptotic approximation
for log Qp (2) vanishes identically.

Therefore we have, provided — tt < 6 < 77, the asymptotic equality

l°g QP (2) = (m ~ i) log - m - 1 log 2 - log 277 + 2' F ( s )

-p S= —p s~ \ P /

7TZP , . Vp'P m 1

+ - - 4- to log — — — 4 log -

1 Sill 7775 & 2 p - * z

Thus, provided — tt < arg 2 < 77, we have finally

* Q. <*> = - i lo§ * - * >°g * + , a (-iF F (7)

This expansion is exactly analogous to the one previously obtained for log Pp (2)
and is to be regarded in the same way. It must be borne in mind that nl,p has been
assumed to he the arithmetic pth root of n. Had any other root been taken — say

27 nr

the arithmetic root multiplied by cj = e - , where r is an integer, we should have

obtained the asymptotic expansion

— (2n r)~*p

Qp (*) = e p exP

7 tz

L sin it p

+ 2

s= —p

SZs

F In¬

valid, when — 77 < 6 — r — < 7 r, i.e., when — 7r + < d < 7 r +

P P P

The expansion is thus valid everywhere except along the new line of zeros.

-7 TV

7 TV

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

471

§ 6G. We proceed now to investigate the asymptotic expansion for

30

F (s) = n

n= 1

+

panP

00 1 00 1

where ctn is such a function of n, <f> (n), let us say, that S — ^ - converges, and 2 — —

n=i b=i nn

diverges, however small e may be, p being real, finite, positive, non-integral, and
greater than unity, while is an integer such that p + 1 > p > p.

Suppose that the result of reversing the equality r — <$> (n) is to give n — ip (o').

Let on be a very large integer, such that on — 1 < xfi ( j z | ) < rn.

As previously, we have

log F (z) = (on — 1) lo

VI — 1

_ s'

n=l

log <f> (n)

+

+

wi-i

v

n=i

'(f) (n)

+

+

(-r1

VI — 1

s'

n- 1

Z ( — )PzP

- - — —

cf>(n) p (/>?<»_

+ 2

n=m

(—)PzP+1

_(p + l)^+I(n)

+

( — )/,+1^P+3
p+2<f)p+2(n)

Substitute now the arithmetically asymptotic approximations given by the
Maclaurin sum formula, and we have

log F (2) = (on - 1) log

"Ml

log (f) (n) dn + \ log <j) (on)

yo

+

/.■

8=-k

(-)- 1

in k

<f)’ (n) dn — \ 5/

y, S=-k

(~y 1 a (m)

sz 4

+

CO

t=0

(_-y a+1
(2 £ + 2) !

log <i> (»o +

S2"

cPt + l
dm~t+l

ft M

In this expansion y_s is infinite, and there is no corresponding Maclaurin constant
if, and only if, s > p.

Use indefinite integrals and transform by integrating by parts in the same way
and under the same restrictions as in § 54, and we get

log F (2) = — i log 2 +

(yo

log (f> (n) dn + S'

5= ~P

( ~)s

sz*

f ft ft1)

dn

ifr (t) dt
t

472

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

T1 le final integral gives rise to the dominant term of the asymptotic expansion.
As formerly, it may be written

Lt | L\fj {z

k=x

sin {k + I>)/4
sin ^ fji

dg.

If we denote the value of this integral by f (2), and if we put

f l°g $ (n) dn = log F0> | cfy (n) dn = F„
we have the final asymptotic equality

co

II

»i=i

exp

/« + *'

S=-p

§ 67. We notice the exact analogy between this expansion and the one previously
obtained in the case when the order of the function is less than unity. The only
difference arises from the Maclaurin constants. In the former case, all the constants
corresponding to negative values of s were zero ; in the present case, the first p of

7/i—l J rwi

them are formed from asymptotic expansions like % — = <f)~s (n) dn + . . . , and

n=l &11 J y—s

give rise consequently to finite constants ; while only the remaining ones, formed

co ^ r m

from expansions like — X — = (71) dn + . . . are such that y_s = co .

n=i)i Q- n •' ' r-s

We notice also the great elegance with which Weierstrass’ exponential factor
enters to ensure the finiteness of the expressions obtained in the course of the
analysis. Could we conceive an attempt to investigate, for functions of order greater
than unity, the theory which we carried out for functions of order less than unity in
the first paragraphs of this part of the present paper, we should at the outset be
forced to invent again Weierstrass’ great theorem.

Application to Functions with Algebraic Sequence of Zeros.

§ 68. We will now evaluate the first few terms of the asymptotic expansion for

CO

, _ (-)PzP !

/ z\ -b - 1 *

1 + \ + h + .

= II

Id - e “* 2 I , where an — n p

, ,

1

_\ ««/

n 1 n -

_

and the

e’s are positive real quantities arranged in ascending order of magnitude.

Let r = an = (f) ( n ), then on reversion of series we find

n = 1 p(r) = - pbp^ + ,*<!-*>> _ pb2f'-^ + . .

When s is positive,

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

473

lil — 1

m- 1

2 a/ = %

n= 1

91= 1

HP + sb^'iP 1 +

el I S (s I) i o — — 2ej

b^np -j- sboUP + . .

/ o \

= j c f)s(n)dn + F ( — ) + sbrF

+ 5^sF

P.

e2 ) + • • •

S

\ P

el +

s(s-l)

b>F - -

- p

*<*>+ I

^ ,fr 21+21 An“+' v ' ’

9

- el , s (s + ] ) , „ - i - 2e,

— p

7/1-1 ^ 7,1-1 r _ S

and X -- = X n ? — sAn / 1 +

*=i «/ 77=1 L 1

= fV(»)rf» + z(P>-»; ££;;;) + f(- p - ^

s ai,i!“ ,,,,

i i0 LU 4 2! ^ W*

where Z i p, — s ; ^ ^ ' J caii be expressed in terms of Riemann £ functions, or the
equivalent Maclaurin constants F by the formula

Z [P> ~ S ; b\ b, ’ ’ '.) ~ “ i96iF (~ p ~ ei ) +

s(.s‘+ 1)

&!2F

- 2en

*&,F - ~ - e3 +

As formerly, we put

z(o: £ £ ; ; ;) = lf (-«,)-£*(- 2*)+ w- %)+•••.

so that log 2-tt + Z(0) arises as the Maclaurin constant corresponding to the

7)7-1

asymptotic expansion for X log <{> (n).

77 = 1

Proceeding exactly as for the case when the order of the function is equal to unity
we see that the asymptotic expansion of log P (z) is

7 r

00 ( — V 1 / <5

- - zp — A log z — - log 27r + X' - F

sin 7 rp ^ & 2 p ^ s=_p sA \ p

TT

sin7r/j(l — ej)

— 7Tp&2

gPd-O 4. dlu + i - 2PU) h2

77

sill 7rp (1 — 2eL)

..p(l-2e,)

• p (1-el

o 7 _ y-i

r?/ V / 13 V

, x ~ + . . . + 2

Sin 7rp (1 — 62) s=-j, 52

p, S

€], fo ■ • .
; &1; 6, . - .

this exp<ansion being valid when — 77 < arg 2 < 77.

3 p

VOL. cxcix. — A.

474

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

Thus, when p — e,p > p, the first term of the expansion of the ratio

jO O'

n: p +;-)«-« .

(~z)P

V«n'

n 1 ( 1 + ~lTP ) * «’/P pnplp

51 = 1 I \ n

IS

7TP\

sin irp (1 — ex)

„0 (l-E))

( _ \Pzp ( 6 , 6n

And, when p — exp < p, the first term is - — — 1 Z i p, —]> ] jV ^ ' ' ' ) , which is readily

( — z)p 00

seen to he equal to - 2

I> ,1=1

_ n

ccj

n

p/p

J'

§ 69. The expansion which we have obtained is valid for all points except those near
the line of zeros of the function and for all finite values of the quantities b and e such
that an has a finite value. It must he carefully noticed that when any term becomes
infinite through the occurrence of sines of integral multiples of 77, we must revert to
the genesis of that term to find the true form of the expansion. Thus, when
p — pe1~ p, the first term of the ratio just considered is that which arises from

-hp | +

, ( — ty 1 dt

=-i- & 1 t

that is, from

, ) J.. ( — )s (b3+P (m) . . / vl

-h'p f - + 1 tf^r + < ■ - # io§ * w i

p

where the double accent denotes that the terms corresponding to s = 0 and s = — p
are to be omitted from the summation.

<Km)

Put now
the series

= e l\ then, with the argument previously used, we have to sum

- b.pfr (m) j j + + (“ <i+ie)p log <j> (on) j. .

Now, when p is not integral, we have seen that

1 ( — V

+ S' e-

7 r

V

s=-l- S + p

sin it p

provided — 77 < 6 < 77.

Let us put p — p + e, where p is a positive integer and e is very small,
have, retaining only first powers of e,

1 . J„ ( — )sc~si9 , ( — )PcPie

_ -j- V" _

P s=-k S + p

+

77

[I + eid],

so that

i- (-ye-*u

— + Sv -

P s=-k S + ])

( — )"7 76

= ( — )p eP‘dW ,

Then we

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

475

The term which we seek is then the value of

- M>f M(-)'

log 2 = ( — )P+1b1pzP\og:

This is, of course, the term independent of 9 in the expansion of

- 71 -P\

zf+e in

sin 7 r

(p + 9)

ascending powers of 6.

In exactly the same manner, if p — nes (say) is an integer, the corresponding term
of our asymptotic expansion must undergo the same process of evaluation and will
give rise to a logarithmic term. If one of the e’s, say e*, is equal to p, we obtain in
the asymptotic expansion a corresponding logarithmic term

— pbt log 2.

Simple Integral Functions of Finite Integral Order.
§70. We proceed now to consider the standard function

R, (*) = n

n= 1

1 + Ti p ) e"llp

n

+ . . . +

(- zU

pn

P P

, where p is an integer > 1.

Let 2 = re1-6, where r is very large, and let rn he a large integer such that
m — 1 < rp < m.

Then, employing the same process and argument as before,

log R„(z) = S {y + . . . + + . . .

^ m— 1 m—l

+ (m - I) log 2 — - t log n + t 1 i/p

P 11=1 71 — 1 I

+ ...+

_ V

n=m

(-*)p+1 _i_ (~zy+2

] « n 4- 9 ”1 * *

p + 1 * p + 2

Ip + 1 )n~p~ (p + 2 )n~

. (~*)P

pil

P/P

Now, when = 1
P

"N1 m‘/p _

s ym*,p + m,/p log m — . -fi

re=l 71 'p S

(-Y

S \ Wk

p 2(1 \2 1 + 2

p m~

and in accordance with the definition of § 49 we put y = F ( — 1).

If, then, we suitably modify the analysis formerly employed we shall obtain, when

3 P 2

470

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

the limits of the summable divergent series as k tends to infinity are taken, the
arithmetically asymptotic expansion

log Rp (2) = (on — 1) log 2 — -{(on ■— \ ) log on — on + \ log 2 77-}

q (-)-1

+ S' fr-F - v//

S=-p

SZa

In n-r1 on°/p

=-* < & S

+ 1
p

( — '\~p ,

-f - — — log m — ^ S

1 S1' y > _

2 ' q/y s

s=—k *>n'

ons’p

a (-Y_BI+1 _1 1-J20! * (-V*

^ £0 2^ + 2! m~t+l 1 P ^

m'

S/p

' r^+]

PA+1

,x '

s/p

*=1

where the double accent denotes that in the corresponding summation the terms for
which s = 0 and s = — p are to be omitted.

As before, the coefficient of B/+1 vanishes identically.

7c ( _ \s-l # 1/p

The series S' ; ??h/p is equal to log

S= -7.: S2S

It is then only necessary for us to consider the series

!■- (—y~l

S"

s=—k SZS S

m

-+ 1
P

If we }mt t = log m_ , we may write this series in the form

f(t) =

7: ( _ \s-l p»t 7c

■ v." 1 — l — A _ s'." / y-i

( — y-lest(-

«=-*• s + 1 1 S— 7c Vs S + P

Remembering that a summahle divergent series may be differentiated, we find

P 5=-/j S

or

/'(0 + p/(0 = ^ + (-)p-1e^

Therefore

f(t) = Ae~pi + t — - + ( — )p“Re“p/,

where A is a constant of integration.
Now when t — 0,

* ( — y-i

f(t) = S" ( >

s~- l

\

S( + 1

P

7c /I

s" (-y-M-

s= — /■ \S

S + pj ’

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

4 77

and, by putting 6 — 0 in the Fourier’s series, which we considered in the preceding

1 / _ y>— 1

paragraph, we see that this is equal to - + 1 — •

Therefore

/(<) - (-r1^

- + 1

L P

+ t-~
P

so

that

2" (uuT1 m*
^ i + 1

= (-r

■i

m

T . , m] p'

L P

+ log

m1'* 1

+ lo« T - P

Revert now to the asymptotic expansion for log Rp (2).
We find on substitution that

log Ptp (2) = (m — 1) log z — - {{m — i) log m — m + \ log 2n }

P

» ( _ y-i

_p v' 1

*=~P

sz-

F(: +(-)'-'=

m1/p . 1
* P

ml/p h L(-«y, n W,p

+ m ( log — J + v - — log m — log

And thus, when p is an integer, 1 2 1 very large, and — tt < arg 2 < 77,

log Rp (2)

- |log2 + £' (-y^-F l^j - ^ log 2tt + (- 2)p log 2 + (-)p 1 7

.S=-p

2 P

As formerly, this expansion is, in form, independent of the argument of 2.

§ 71. We may easily deduce this theorem independently as the limit of our former
results.

Take the asymptotic equality

n ; (i + ^le

n = l 1

■ + ... +

<-*>*- 1

P~l
2>— 1 11 p

*

® 1

= (2Tr)~ty 2_i esmnp

+

UU / X O - J

2' k±_ p/q

S=-p+l \P/’

where p lies between — 1 and

Put now p = p — e ; then Ri; (2) is the limit, when e vanishes, of

n

n = 1

1 + e

n 1

1

,p ~e

- FA

U-G np-*

X n e*n

rt. = 1

p

p-«

478

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

It therefore possesses an asymptotic expansion which is the limit, when e
vanishes, of

(2") exp jsing(p_e)

TTZ?-*

+ UAp *

V

x / — y— i j s

N' ^ ’ F

+

p-ej s=-p+i sz°

P—e

= (27T)

z

'* exP \ ( - ZY l°g z + ~ 1 1 + ^ f v ^

p

S=z—p + 1

= (2 IT) -is-* exp {(- *)' log * + (-r-’f + F

i(*) + r-

= y + terms which vanish

remembering that F ( — 1 ) = y, and that L t

S=1

when 5=1.

We thus obtain the same asymptotic expansion as in the previous paragraph.

Note that we have obtained our expansion by making p increase up to the nearest
integer. If, on the contrary, we make p decrease down to the nearest integer, there
is no breach of continuity in the introduction of an additional exponential factor.
Thus we have

(z) = Lt n (l -f-

e = 0 n — 1

— — + . . . +
1

,.P+e

pn

V

p+e

n

P+e /

and therefore we have the asymptotic expansion

_ 7 TZP + e

['Zn) z * exp \ -t—

c = 0

(z) = Li (2 7r) 2 pif2 i exp

+ 2

/ (-)

t-i

sin 7r (p + e) 1 S=LP sz f \p + e,

Now, unless s — — 1, F (s) = £ ( — s) ; and therefore we have asymptotically

t) i x T . /„ f(— z)p(l + elog* + . ..) (—z)p ( p

B, (2) = L« (2tt) 2r+. * exp | - - - + — £ [—

+ S' - 'ph F f --X-

= (Shr)-*.- exp {(-:)- log « + f(£

the same expansion as before.

This paragraph is instructive in that it shows how the asymptotic expansion calls
for another exponential factor in each term of Weierstrass’ product as the order
passes through an integral value.

§ 72. If now it is desired to construct a function which is the natural extension
among simple integral functions of the ordinary gamma function, we take

r (zip)

= e 5=1 *

s F(~i) A I /. . 2 \ nr+ • < • +

I1 + 5®le ”

<-*f

Up

pH

p/p

. MR, E. W. BARNES ON INTEGRAL FUNCTIONS.

479

And now the asymptotic expansion of T (z | p) when \z\ is very large and
■ 77 < arg 2 < 77 is given by

(2tt)2p z

_L (-)P-1 ,-p-A 2 ~r F (— ) +

2p 7 ' 2 p s = l szs ' p ' P

When p — 1, this formula is exactly the asymptotic expansion of T (z) for complex
values of z, which, as stated in § 3, was first obtained by Stieltjes.

For, when s is an even positive integer, F (.s) = 0.

When s is an odd positive integer = 2t 4- 1, let us say, F (s) = * yp1 . t > 0.
And F (— 1) = y.

® 7 _ y / s

Thus, when p = 1, T (z ' p) becomes T ( z ), and the series 2 — ~ F ( -

S = 1 SZ° \ P

becomes

(-y-Bi+1

tZo 2t + 1 . 2t + 2 ' z2t+ 1

, which accords with the usual result.

§ 73. It is obvious that we can now at once write down the asymptotic expansion

for G (z) = II

7, . z \ -A+... + W'

I 1 + — e a“ “”p
\ a„ 1

, where a„ = n?

1 -L -1- + ^ + .

1 01 €\ ' 01 e 2 * * *

and

p is an integer, from the corresponding expansion for the function in which

1 + + • • •

and p is not integral. The e’s, of course, are assumed to be

positive and in ascending order of magnitude.
The result is

log IT

1 +

+ . . . +

{-)*zV

O-nV

= (~)v zJ log z + (-)

ZP

V

log

2p lo§' 2?r +

00

s= —p

(-)1

szs

i / \v — 1 - p?-*\v

' sin 7T|?e1

. / P(P + 1 ~ , a _ ZL

"I" \ ~~ ) 9 °1 oin 9m

sm Z7rpe1

_1_ ( __ )P - 7p-e2p I I Z' (_ 1 7 I v g. eV e2- ■

+ ( > sin irpe* * + ‘ + ,=-P ** \P’ ’ K h . .

Zip, 0 ;

provided 7y, ^ ... be not integral (ft = 1, 2, . . . co ).

Thus, e, not being integral, the first term of the asymptotic expansion of the
quotient

is

( -- zv Z (p, p ;

We note that Z

61> e2> • • A _ J

ij, . . .) n = 1

' 1

aJ

T

71

480

MR. E. YT. BARNES ON INTEGRAL FUNCTIONS.

When peY is an integer, we see on evaluating the limit which arises, that the
dominant term of the asymptotic expansion is still the one just written down. For,
in this case, the only other term which might he considered first in the asymptotic
expansion of the quotient is ( — )i>+l ph1 zv~€ll> log £, which, since is positive, is of
lower order than

( — )p vry( 6,,

z? Z p, p ;

P \ bV °2>

§ 74. It is now evident that, if we are given any simple function of finite integral
order, we can find its asymptotic expansion. The analysis just given solves
completely the case of algebraical zeros. When the zeros are not algebraic we may,
and, in fact, we shall have to introduce new analytical functions defined as indefinite
integrals ; hut there will be no essential difference in the theory.

It should be noticed that just as we have to take the principal values of the
algebraically many-valued expressions which occur in the asymptotic approximation
for functions of non-integral order, so we must assign principal values to the
logarithms which occur when the functions are of integral order.

Part IV.

The Asymptotic Expansion of Repeated Integral Functions.

§ 75. As has been stated in the general classification of Part I., an integral
function, which is such that its nth zero is repeated a number of times dependent
upon 7i, is called a repeated function.

If the number of sequences of zeros be not infinite, the function is called a simple
repeated function ; and it is obvious that such a function may be built up of
functions, each of which possesses a single sequence of zeros. We shall limit
ourselves to the consideration of such functions. The order of simple repeated
functions with a single sequence of zeros has been previously defined. Taking this
definition, we consider, in turn, in the ensuing paragraphs, functions

(1) of finite (non-zero or zero) order less than unity,

(2) of finite non-integral order greater than unity,

(3) of finite integral order greater than or equal to unity.

And, finally, an example is given of the asymptotic expansion of a repeated
function with a transcendental index.

Inasmuch as the principles which underlie the analysis are exactly the same as
those which have been previously discussed, we shall give but a bare outline of the
methods by which the results are obtained.

ME. E. AY. BARNES ON INTEGRAL FUNCTIONS.

481

Simple Repeated Functions of Finite Order less than Unity.
§ 76. The most general function of this type may be written

F (z) = n

?i=i

1 + a.

where the principal value of each term is taken when y„ is any function of n ; and
where an is a function of n which increases without limit as n increases, and which is

oo

such that 2 yfa,, is absolutely convergent. The function is of finite (or zero) order

n= 1

less than unity ; and, when yn is an integer, its nth zero is repeated y„ times.

We take

an = (f) ( n ).

When r = <£ ( n ) we suppose that inversely n = */;(/'). Suppose that z = Re'9, then
if we take m to be a large integer such that m — 1 < \ft (R) < m, we have

m— 1

m— 1

iii — 1

n= 1

n= 1

n = 1

log F (z) = 2 y„ log z — 2 y„ log <£ (n) + 2 y„ log ( 1 -f - ) + 2 y„ log ( 1 +

a.

We carry out our analysis in a manner which depends exactly upon the argument
previously employed in the corresponding case for n on-repeated functions.

We have at once, in the limit when h — «,

rn— 1

m— 1

n tt1 / \ i v ~\ j/\, ’V i (— y iy„F(n) , o k (— y

log F (z) = log z 2 yH - 2 y„ log (n) + 2_ 2 - — - + 2 2 s.^(/,) •

11 = ] s=l

n=m s=l

Now, if .s be positive,

in— 1 pth

2^ y„(f>3 (n) = I y„f(n) dn - byj/ (m) ... + (-) ^ + 2 ,

Bm d2<+1 . .

- ym<f>s(m) +

where ys is a constant depending on s and on the forms of yn and <f> ( n ).

We call ys the .sth Maclaurin integral limit for ya and 4> (n). If s be negative, the
previous expansion will hold, but in this case y_y = oo , and the constant term vanishes.
Again we have

m— J rrn

2 y„ log (n) = | ytl log <f> (n) dn — \ ym log <£ (m) + . . .

n= l •'■yo

+ (-y

B<+1 .4P*

2t + 2 !

ym log <f> (m) + . .

and

-1 rm g<+ pt+l

P>i — | y-n dn ^-/xw + . . . + ( ) 2^ _|_ 2 ! dmit+1 " ‘

3 Q

VOL. CXCIX. — A.

48-2

MR. E. W. BARXES OX INTEGRAL FtTXCTIOXS.

We shall find it convenient to put

rr/o

/ u,„rhi = x(m)’ so that I jj.„dn — x(ffo) = ~ M (say).

And now, if the limiting values when k = =o of the summahle divergent series he

' o o

taken,

log F (z) — log z

| jjL„ dn d- dl

"Til

log <f> (n) dn +

^ Vo

m

(n) dn — \ ( m )

7'

^■S

c-~

rt

=£

f^?n

_!y_.5 <ps'0d

2 A(m)_

+

(-y

r»/+l

f72/+1

f=0

'it + 2 ! dm~t+l

/x,„ log (p (m) — fim log z +

/.

v

S=1

(— y

jwhl
A Cm) 1

The last term vanishes as for the corresponding case of non -repeated functions.
After reduction, we have

log F

= loc

Cm “| cm

I jjL„ dn A M —| ji, log (f) (n) dn

' 7o

i- ( _ v-i r r m ,, i

+ S { f'" M (/,) dn -

*=i ■<* [Jyt A J y_s & (»

(d

d/z

= M log z A fjt,, log r/> (77) dn + 2

t (-)s r*‘

=1 sr

(n) dn

rMm) x [A(0] dt

The last integral

A, (— yt*

1 A 2 -

= U f x [* ( - **)] i # ?n- Ay ' * = /<*) say.

If then we put log F0 -= n„ log <f> (n ) c/77, F.v = — | (n) dn, so that F0 and

Fj may he called the zero and sth Maclaurin constants for /x„ and (f>(n), we shall have
the asymptotic approximation

n 1(1 A -

*= 1 L\ Oh

\P«

/(-")+ i

(-)s-h

= FnSMe S=1

MR. E. AY. BARNES ON INTEGRAL FUNCTIONS.

483

§ 77. Consider now, as an application of the general formula just obtained, the

asymptotic expansion of IT

>1 = 1

1 + ,p

, where <x and p are real positive quantities

■jj

such that % ua~p is convergent, and, therefore, such that p > a + 1.

71=1

With our former notation

p, = n",

rm

X (m) = ] dn
' !'(t) = t1/p.

m

<r+ 1

CT + 1

X ['A (0] =

a + 1

t~T~

CT + 1

The constant <j{) arises from the asymptotic equality

iil — 1

v

n= 1

nu =

x (z&i ^+1
,r0 (2« + 2)! dm~lPl

and, therefore, M = — x (g0) = £ ( — <r).

Similarly j pn ft (n) dn = — £ ( — ps + <x).

The constant y0 arises from the asymptotic equality

//i-i

5 na

ii=i

loo- n dn —

O

Vo

m

9

log ?n +

y (-J9T+1

tio 2* + 2! dm***1

(m0- log in ).

ry o

We may readily show that n* log n dn = £' (— <x).
For, as has been stated, for all values of 6',

m-1 1

£W = s i

7( = l /t

— s ?/i'

y— 1 +

+ t

t- 1

— s

2t

( y-iB<

(s + 2*

1) m*+s<_1

If, then, we put 6- = cr + t, and expand each term in powers of t, we may equate
coefficients of similar powers in the identity. #

If we equate coefficients of the first power, we find

Ho-)

m~ 1 loo- n
V _» _

a

n=l 11

m

L(i - o-y

log VI

1 — a

log m -F

v (~ fWi cI2t+1 Ipg 711

t-o 2^ + 2! dm-t+1 rri

* Compare the process carried out in §§ 27 and 30 of the “ Theory of the Gamma Function.”

3 q 2

484

ME. E. W BAEXES OX IXTEG-EAL FUXCTIOXS.

or, changing cr into — oq,

m — I

S na log

n=l

rrn cr CC ( — VR, fPl+l

i « = j ^ log « dn - i'(- <r) - 2 log m + s ^TT «i<T log m-

Thus

Cy o

717 log = £' ( —

cr 1

We have, therefore,

log n ( 1 + ;y) = £(— a-) log 2 + pC{~ o-) + 2 ( ~~ l ( - ps + cr)

.,= 1

<t> M

]_ f 7 <7 + 1 <7+1

, 1 f * <

+ 7TlJ 2

p t~p 1 clt

1 + X' ( — )T

s = -k

The last integral is equal to

<T+ 1

[<f> (m)] p

P _J_ 4f J

" - 4>(m)

4 1

cr + 1

, + iTii

_ [

[ * J

<j+ 1

<b (m)]p

cr A 1
s +

p _

_ 77

cr+ 1 :r + l

P IT Z P

cr + 1

. 7T . C7 -f 1
Sill -

_<f> (m)_

cr + 1 cr + 1

Sill 7T .

Thus we have the asymptotic expansion

II

n=l

7 , ^

lV + ,f

<T + 1

7 T -

s , - z p

?£(,-*) »?£(-*) + - f(-p« + <7) + sin :i+T+1 0-+1

z e *=1 p

We note that the first term of this product vanishes when cr is an even integer.
§78. It is now possible to write down the expansion of

where \xa is algebraic and of the form a0 n'T + cq na' + a.z na- + . . . , in which
cr > cr1 > (T.i > ...

For such a function is merely the product of the

CO

«0th power of II

n- 1

the cqth power of II

H = 1

and so on,

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

485

We note that the constants which enter will be expressible in terms of the
coefficients of p„ and values of the Riemann £ function.

We might now investigate the asymptotic expansion of a repeated function of
finite order less than unity with algebraic sequence of zeros of the type

a» = np

A &.•>

'+h+?+

where the quantities els e3, . . . are real, positive, and in ascending order of magnitude.

The analysis is, however, such an obvious extension of the corresponding result of
Part III. that it may be at once supplied by the reader.

Repeated Simple Functions of Finite Non-integral Order greater than Unity.
§79. We next consider the asymptotic expansion of the function

F (*) = n

n=l

, , z Y" % l/z v>

1 + — ) en mhj m (" a)

'd'n /

where p < p < p + 1 , and p is such that

2 is convergent, and 2 divergent,

CCn

/J'n

when e is a small real positive quantity.

The analysis is an obvious modification of that employed in § 66.

We find F (z) = F0 zM e/(f)+,*Lp N , where an = <f> (n), y (m) = | p„ dn,

in-1 rm

2 dn + M — pm + . . . ,

51 = 1 j

5JI—1 Cm

2 Pa log (f) (n) = P„ log <f> (n) dn — log F0 — \ ym log <f> (m) + . . . ,

n = 1 J

771—1

71 = 1

5 - X Uly

2 p„fs(n)= p„ dn + F,— -f . . . (s=—p, — (p— 1), . . . — 1 , 1, 2, . . . oo ),

and

f(z) = Lt i f x[i \j (-zef]

k= x> J

sin (7j + j)
sin I <j>

Icf).

§ 80. As an example, we may consider the function

CT+ 1

F (z) = n

71=1

, where is not integral, and p < __ <_p + 1,

The order of the function is — - — - .

486

ME. E. W. BARXES OX IXTEGEAL FUXCTIOXS.

We have

log F0

- X (9o) = £(— o"-

s ii~ 1 r m rj'

X t na log ;/ -j- | t n° log n dn — - n<r log /< + ..■= r V, l cr),

«=i

— F, = — £ ( — t S + cr).

»T T*

And, as in § 77, f(z) =

+ 1

. 7r (cr + 1) (cr A 1)
Sill -

Thus the asymptotic expansion of F (2) may be written

-A(-<r) e sin-(?Ai)

<r + l

(-)- ]?(-<T- rs)

8= —p $ Z

Note that

l (0)# = -i, t (0)f = - l log 277.

1

Hence, when cr = 0, r = -, we get the asymptotic expansion

n

n= 1

L + 1 <2

MP

ii=l m V n

— Z * e sin - p ' 2p log 2 7T +

(-)" H( p)

*=~P

which agrees with the expansion of § 65.

Simple Repeated Functions of Finite Integral Order.

§81. It is obvious from the investigations of §§ 70-73 that the asymptotic expansion

, where p < p < p + 1 and p is such

that X is convergent, and X divergent, will hold in the limit when p — p,

-■n G>n

provided that in any terms which become infinite we reject the infinite part and
keep only the corresponding finite expression found by applying the usual methods of
the calculus of limits to the subsidiary Fourier and Maclaurin series. Consider, for
example, the function

obtained in § 70 for II

n= 1

/ 2 \P» V 1 / r. \„

1 + -

\ ]

II

,' = 1

where

(7-4-1

The asymptotic expansion obtained previously was

* “ Theory of the Gamma Function,” § 27,
t Ibid., § 30.

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

487

<T+ l

:■{(-*) o sin

7r(a-+l) U+<T)

+ T<J"(— O') +

?/ (~)'S *£( — cr- T.s)

s=—p szs

Now, when — + - = j) = an integer,

7T

<X + 1

Sill

(o- + 1)

« x becomes infinite.

We have then to substitute a corresponding finite expression derived by proceedin,

1 .

to the limit in the infinite terms of the subsidiary Fourier series. When is not
integral, the series and its equivalent value are given by

+

•S'T

1+CT

+

( — )sesW

ire

•=' Wi

l+o-

sm 7 t

<T + 1 1 +

id.

When - is integral, this series, omitting the term for which .s' = 1 + in the second

T T

summation, is equal to the finite part of

cr+ 1

T (1 + ei6 + . . .)
1+0"

( — ) t e

1 + cr

— id —

1 +cr

T(-W jg — 1
— - - e t

1+0-

when e = 0 ; that is to say, it is equal to

(-)

s

We thus replace

+ 1 , _ <r+ 1 / . _ t

’ e* v ( - 16 +

1+0-/ 1 + 0-

iO.

cr+ 1

7T

sin

7T (o- + ] ) 0 + 1

'T+ 1

i (— *)~ Jb

by , {log2 -

1 + 0"

Again. since <x is a positive integer, the only term of the form

£ (— cr — ts) s = — p, . . . —1,1,2,...

which becomes infinite is that for which s == — p.

This term is £ (+ 1), which arises from the Maclaurin series

m— 1 ^
£ _

»=i n

{(») +

1 — s ms

— — — h • • • = log m + y — - — b

S=1 2 in & 1 / 2m *

We have already taken account of the substitution of log m for _ . — 1 ; we need,

therefore, only replace £(fi- 1) by y.

We have then the asymptotic expansion

488

ME. E. AY. BAEXES OX IXTEGEAL FUXCTIOXS.

", II1 +7)"

<7-f 1

= W <r1 exp

rr (- o-) + “f~ {log j + 7T - f£

+

w Y_ 1 — Tf)'

J=l-

<T+1

sz*

in which — and cr are hoth integers, and J + is the “ genre ” of the function.

It is interesting to notice that the constants which enter into the asymptotic
expansion of this very general function are all values of the Riemann £ function.

§ 82. When r = 1, the function is to an exponential factor an important function
which I have proposed to call the cr-ple G function. These G functions are derived
from the multiple Gamma functions by the coalescence of the parameters. The theory
of the simple G function has been developed elsewhere* in the second of a series of
papers on Gamma functions.

In that development I took

Gr (z -b 1) = (2tt)2 e

Z . 2d* 1

n

'<1=1

{ 1 + - ) e J+ 2»
v n

and obtained! the asymptotic expansion

log G (s + 1) — ra — log A + 2 log 27r T ~ rVj l°g 2 ~ 4 + ^ 2772^+^2 z2*’

where A is the Glaisher-Kinkelin constant.

Putting cr = t = 1, the asymptotic expansion which we have obtained for the
same function in the present paragraph is

I log 2. - *Y+i) _ y + {(_x) iogz + 1 log, + y _

Now J

and

+ £(- 1) +

£(-i)= -A. £(o)=-i.

£ ( — .<f — l) = 0, when s is odd,

G (-r1 si-*- 1)

.<=-i

s~

1 B

(2 1 + 2)’ when * =

* 1 Quarterly Journal of Mathematics/ vol. 31, pp. 264 et seq.
t Ibid., § 15. + Ibid., § 23.

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

489

And, since A is given by the identity,

in — 1

2 n log n = log A +

n = 1

m -

ITt \ r¥T$>

— ~ -f- j A j log m — — + terms which vanish when m is

infinite, we have log A — — £' ( — I) + yy-

Thus the asymptotic expansion of the present paragraph may he written

T2 - log A + * log 2tt + -~h) log 2 ~ y

+

(-)SB,+1

s=i 2s . 2s + 2 23s’

We thus obtain a valuable verification of our results.

Repeated Simple Integral Functions of Transcendental Index.

83. It is obvious that such a function as

n

k=i

is of infinite order when an is algebraic.

If, however, an is of the same order as en, tlie order of the function is finite, and can
therefore be expanded in the neighbourhood of infinity by our methods.

We shall take, as an example of repeated functions of transcendental index, the
product '

GO

n

71 =1

&n

nn £

£ /IV'

,=A «v

This function is of order p, greater than or equal to unity.

Suppose first that p is not an integer, so that p is the integer next greater than p.
Then without former notation

VI — 1

^ eMn — •

ii =i

ra— I

2 emn qn =

v. = 1

fm enn

| emn dn + M — — + ...

fm

evqn qn dn — log F0 — \ emm qm + . . .

And

m— 1 rr,

2 q'M'1 + s2'! —
rt = 1 J

l pp+s qm

e(p + s)an dn _Ys — ~ - - + . . .

r , . log z m ,1 , I. 1 f( — )-s_1 e?+

f (z) = — — ePf?W! — — eJ'im A - epim + 2

' m P FI .=-t2 L

5 qm

p + s

where we take the limit when k — co of the summable divergent series.
VOL. CXCIX,- — A 3 R

490

MR, E. W. BARNES ON INTEGRAL FUNCTIONS.

Thus

/M _ Io§ g _

1 £ pqm q

1 1 *
m + + - t'

PI I =-*

'( — )’-1 /g?M

1 4' r(-)8-1

I s=-Jc _ P + s \ 2 . J

log 2 | 1 | 1 J 7T ,

— m — + - i /

2

PI

Z \P

Sill 7 Tp 'P

— 1 — log —

Therefore

Again, M is given by

/(*) =

77 zr

pq sin 7 rp

a am

e — G

1 - e“

p + Lt +

r=co «

l-o +

/,• ( vn ,,2;+2'

^ (, j Of+i <*

<=0

2^ + 2!

• • (1),

dm

e a

= M -(- — . - - - , when a = pq.

a 1 _ e 11

Tlius

M =

eji

1 - en

Also — Fj is given by putting a = p + 5 q in this same expansion.

Thus

F —

e? . r+-!

1 — eq _ p + s

Again, by substituting a + e for a, expanding in power of e, and equating
coefficients of the first power of e in the asymptotic identity (1), we readily find

evi

— log F„ = q — - — .

& 0 1 (1 — e^y

If, then, p is not an integer, we have the asymptotic expansion

n

n = 1

i + A

1 anj

. rpqn vnn . „ .

e ' 1 cl 1 p ( —z \m

m= 1

(w

V'l

= z Qpq sin irp (l-e**)*’'' MjLp T7).<

§ 84. Suppose next that p is an integer — so that p = p. The analysis will, of
course, he slightly more complicated.

The constant F_„ will be given by

m— 1

Lt \ S j _ + dn _ F_p _ ]

s=p L »=i j J

or

m — l = m — \ — F_p, so that F_;, = i

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

491

And we shall have
pf 0) l°gz ■ 1

mzP

CM”

C1

pq ’ ep«m 2 s=-7. L s \ z J J 1 *=-£• V.P + s

_ 1 f-'

s = 0

the double dash denoting that the terms for which ^ are to be omitted.

We therefore have

pff) ]°g

tfqm

- -_m +

q pq zn“l pq \c?i"

i

t z \P

Therefore

+ q l0S \ffn]

f(z) = — ■ - zp log z — {^~zp .

J \ ' pq & p-q

\ 1

P

- lo

O'

We thus have the asymptotic expansion

n

>i=l

epqn

1 + -) e*m
ci'1 j

= z (1_c n ) le

(,-z)P\ogz (~z)p

M

P<1

pH ?

(i-uA

o + '

c-W1**

ip

X <?S=-/J + 1 s(l-f2(2J + «)),«

We have now given examples of the asymptotic expansions of repeated simple
functions with transcendental index in the cases when the order is or is not integral.

And it is evident that such examples might be multiplied indefinitely. In the
more complex cases the difficulties of the analysis will, no doubt, be very great ; but
such difficulties in no way invalidate the theory which has been developed.

Part Y.

Applications of the Previous Asymptotic Expansions.

§ 85. We proceed now to consider some applications of the previous theorems to
such questions concerning integral functions as have been raised in the Introduction
to the present paper.

In the first place, a knowledge of the asymptotic expansion of a function serves to
determine the number of roots which it possesses inside a circle of given large radius.

Let us consider the simple example of the Gamma function, for which we have the
asymptotic equality

1

PW

= (2-)-^

1 exP \r

3 R

+

(-)‘B

V+1

1

,=o 2* + 1.2* + 2 2:

> -27+1

492

MB, E. W. BARNES ON INTEGRAL FUNCTIONS.

in which the terms neglected on the right-hand side are of lower exponential order
than those retained.

By Cauchy’s theorem the number of roots N within a circle of given large radius
r is determined by

N = - 1 ( y log T (z) dz,

2m ) dz & w ’

the integral being taken round the circle in question.

Now we may, to terms which vanish exponentially with ^ , substitute for T (z) its

value given by the asymptotic expansion. And this expansion is valid for all values
of z for which — n < arg z < v. It is also valid right up to the two limits of arg 2,

provided the circle on which z lies passes between two consecutive zeros of •

If, now, 2 = r&°, we have, to terms which vanish exponentially with y >

log r — lO — y +

2 (~)'B,+1 c-

t=o 2^ + 2 r-i+-

re

cie.

Now

ir u n a in ~ U'

0e‘e cie =

LIT

LIT

- 6<r‘6

L

7T 1 [n

dO

— r

— inel* + m

0—iir 1

= V.

Therefore, to terms which are ultimately exponentially small

N = r -f i.

Of course we know independently that
the number of roots is the greatest integer
less than r. And the entrance of the
term \ might have been predicted a priori,
for when the circle of radius r passes

through a zero of ^ - we jump from —

to + as we integrate round a small
circle enclosing' this zero.

8G. It is interesting to notice that the analysis verifies itself in the same way for

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

493

the function F (z) = IT

n = 1

1 +

en

, which, by a change of the independent variable,

reduces to Lambert’s function.

For this function we have obtained the asymptotic expansion

log F (z) = 1 (log zf ~ i log 2 +

t r2 * (-r1 0,

6 1 si, S3'

Therefore y log F (z) = ^ - 1 +5 (— ^

dz & v ’ z 2z T s=i z*+l

The number of roots within a circle of radius r is, therefore, to terms which are
exponentially small when r is large

2-7TC

log r

e~'6 + — {l6 - i) re 10 idd = log r - 1.

Since the function has no zero at the origin, we should ha.ve predicted the occurrence
of the term —

§ 87. We may now prove that, if the dominant term of the zero of an integral
function is algebraic and such that the zero is of non-integral order . p (where p is
neither zero nor infinite, but greater or less than unity), then the number of roots of
the function within a circle of large radius r is to a first approximation

sill 7 Tp

IT

log <f> (r),

where <£ (r) is the maximum value of the modulus of the function on the circle in
question.

There are two cases to be considered according as p is greater or less than unity.
We take the former, the argument will hold in detail for the latter by changing p

into — .

9

Let F (z) be the function in question ; then, under the conditions enunciated,

7T

log F (z) is equal to . - zp - f- terms of lower order.

Hence N, the number of roots required is, to a first approximation, given by

N =

rp epL(> l dO =

rp

sin 7 rp

IT

log (r).

We thus complete and prove Borel’s intuition.

§ 88. When p is an integer, the preceding theorem ceases to be valid. But we can
now prove that the number of roots to a first approximation is — ' .

494

MR. F. W. BARNES ON INTEGRAL FUNCTIONS.

For, the same conditions still being supposed to hold good, log F ( z ) has to a first
approximation been proved to be equal to

( — z)p log z + ( — )p 1 ' + - — - - z? -f lower terms

And therefore

(l

dz

log F (z) = ( — y pzp 1 log z -fi lower terms.

Hence, to a first approximation,

N = j" p ( — y ein6 rp [log r -f 16] idO

log 0 (r)

= ( - & f-

A 77

elu9 0 dd — rp —

log r

which establishes the theorem in question.

§ 89. In the two preceding paragraphs we have assumed that we were dealing
with lion-repeated functions.

From the analysis of Part IV. it is, however, evident that the theorems hold
in toto for repeated functions, the order being that which has been assigned to such
functions.

We cannot, of course, attempt to prove the theorems for functions of multiple
sequence until we have investigated the corresponding asymptotic expansions.

§ 90. We may next write down a number of theorems relating to two or more
integral functions.

It is obvious that the sum of two or more integral functions of simple sequence is
an integral function of order equal to the largest order of the component Integra^
functions. We may replace additive signs by those of subtraction if the two
component functions of largest order are not identically equal. The large zeros of
the compound expression are to the first order of approximation equal to the large
corresponding zeros of the component function of largest order.

The product of two or more integral functions of simple sequence is an integral
function of order equal to the largest order of the component integral functions.

The number of zeros of the equation

/q (z) F, (z) + h, (z) F3 (z) + • • • + K (z) F. (z) = 0,

where the F’s are integral functions of simple sequence and the h’s algebraic
polynomials, within a circle of large radius is ultimately to a first approximation
equal to the number of zeros within that circle of the function of largest order.

§ 91. The expansions which have been obtained may be utilised to give a proof of
Bokel’s extension of a theorem due to Picard A

* Picard, ‘ Annales de l’Ecole Normale Superieure,’ 2 ger., t. 9 (1880). Borel, ‘Acta Mathematica,’
t. 20, pp. 382-388.

mr. e. w. Barnes on integral functions.

495

The identity

n

S'

*-4

t = l

G, (z) = 0,

in which the G’s are integral functions of simple sequence of any finite or zero order
not greater than some number p, and the functions Ht — H,, are polynomials of order
greater than p or transcendental integral functions, necessarily involves

G, (: z ) = G.3 (z) =

= G u(z) = 0.

Since the identity holds for all points in the plane of the complex variable 2, we
may consider it in the neighbourhood of 2 = <x> .

Suppose first that the G’s are functions of simple sequence of non-integral finite
order.

If pL be the order of Gt (2), we shall have near 2 = 00 the identity

X e sin

1=1

,pi

H (2)

= 0,

where we have neglected in each term terms 01 lower exponential order than those
retained.

The identity will hold -for all values of arg 2 such that 2 is not within a finite
distance of the zeros of the G’s.

The functions H (2) by hypothesis cannot be equal to one another. As 2 tends to
infinity, one of them must become infinite to an order which exceeds the order to
which all the others become infinite by a quantity of order greater than zp.

The corresponding term (say) G] (2) eH'fz) is then infinite to an order greater than

n

the order of any other term of the identity X Gt (2) eH‘(2) = 0.

t=i

Since eH(:) cannot vanish, we must then have Gx (2) = 0.

n

The same argument may now be applied to the identity X G( (2) elI‘(s) = 0, and it

i=2

may be proved successively that all the functions G vanish.

And thus the theorem will be proved.

When any of the quantities ' p are integral, a suitable modification of the formulae
in accordance with §73 shows that the theorem is still true. When the G’s are
repeated functions, a corresponding modification again establishes the theorem.
When the functions G’s reduce to constants c„ so that p = 0, the theorem is still true,
the functions H being unequal. .

§ 92. We pass now to the consideration of the resemblance between an integral
function of simple sequence and its derivative.

And I would remark that, in the same manner as Rolle’s theorem is proved, it
may be established that the real zeros of such a function with real coefficients are

496

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

separated by zeros of its derivative. [It cannot, however, be proved that the derived
function has not other real or a (necessarily even) number of other imaginary zeros.]
This theorem I shall not prove, as it is not connected with the main developments of
the present paper. We proceed, however, to show that such developments complete
and to some extent verify this extension of Polle’s theorem, and that incidentally
they furnish many criteria as to the nature of the derivative of a given integral
function.

93. Let us consider, as an elementary example, the function of genre zero and

order — ,
P

P„ (*) = n

',i=l

1 +

IV

, where p > 1.

We have the asymptotic equality

Pp (z) = (2t t) 2 z 1 exp.

7 r

sin

7 r

ZP d~

( -y-'Fipsi

Pemember, now, that it has been proved in Part II. that we may differentiate an
asymptotic equality of this type, and we obtain

frpp(2) = (270 22 -exP

^ zl + . . .

7 T
Sill -

P

7T

I=£ 1

Z p — -p . . .

7 r pv '

Sill -

t P

J

x_P 7T l±p | 7 T

= (27r) 2 u ^ 2 p exP

ZP

+

together with terms whose ratio to the terms

sm -

sm

P L p

retained tends to zero as | z | tends to infinity.

From this expansion we see that

(1) Pp' (z) is of the same order as Pp (2),

(2) The zeros of P/ (z) are such that, when n is large, we have with the usua

notation an = np + (p — l)?ip-1 + lower terms.

Not only so, but theoretically, by finding successive terms in the expansion for
Pp' (2), we ought to be able to determine the form of its nih zero as nearly as we
please. Practical difficulties will, of course, arise when we come, in the asymptotic
expansion, to a term which arises from a transcendental term in the nth zero of Pp' (2).
Note that the formula for au may be readily verified when p — z.

For

and, therefore, P3' (2) =

n

n=\

sinh 7 T\/z
7T\/ Z

jL cosh 7 r^/z x sinh 7r z

T*S Z

— ^ _.,/;2 asymptotically.

z

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

497

The large zeros of P/ (2) are approximately those of cosh 77 v 2, and the latter are
such that, with the usual notation, a„ = n2 + n +

We notice that the general form of an given above shows that the large zeros of
Pp (2) separate and are separated by those of P/ (2), a fact which agrees with the
extension of Rolle’s theorem.

§ 94. From the preceding example it is now evident that we are in a position to
prove that, for all the types of integral functions of which asymptotic expansions
have been obtained in this memoir, the order of the function is equal to the order of
its derivative. And not only so, but we are theoretically in a position to determine
as nearly as we please a formula for the (large) nth zero of the derivative. It would
be tedious to consider in turn all cases which can arise.: we will take one or two as
typical of the rest.

As an immediate corollary of the preceding example it may be seen that the

derivative of a simple non-repeated function of order - less than unity with algebraic

zeros of the type an = np -f 9np~l + . . . is a similar function of equal order, whose
zeros are typified by ba = np + (9 -j- p — I) np~l + . . .

§ 95. As a suggestion of the possibility of extending the expansions of Parts III.
and IV. let us next write down the first few terms of the asymptotic expansion of

P ( a + 2) = log n

1 +

z + a\ - z_±£ + . . . + <-)pC+«)p-|

an

6 an

pani‘

, where a is any quantity of finite

modulus, and an = nl,p

h

1 17/ 1 1 1l€2 '

The expansion will be (§ 68)

77*

00 ( _ V ^

- (z + a)p — \ log (2 + a) — — log 277 + S' -7 - .. ^ , ,

SUIT rp v 1 ' 2 &v ’ 2 P * ,=-ps(z + ay \pj

_ _ _ u _I_ a)p (1-6l) 4- p (p + 1 ~ 2p€l) b 3 _ _

sin 71-p (1 — ex) ' ' 2 1 sin irp (1 —26])

F ( —

V P /

(2 + a)p (1_2eA

sin 7 rpK , », ( — )s_1 „ ( €,, e9 . .\

-7 - 77-^ — - (2 + a)p(1_e2)+ . . . + S -7 - r Z Ip, s' J1 j

sm 7 rp (1 - es) V ’ s=-ps(z + a)s \ \ . . ./

— Z ( 0 ;

6], e3 . . .

\ ’ Kh---I ’

and may be transformed into

77* 1 go ( — V 1 fS

zp — ^ log 2 — — log 27t + S' - — F

5=— J)

sm7r/3 " 0 2p ~n " ' szs

+ 2P_1

+

P(P~ 1)7r
sin 77 p

V -

aaz'

p Sill 77 p

+ (-)■’ AAa2F(A + (-)>'-IaF(£-^) z--1

+ • • •

[Over.

VOL. CXCIX. - A.

3 s

498

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

^ 1

+

sin 77 -p (1 — e2)

pip + 1 ~ 2<h)

2

irpb 2

yP(l-el)

7rp25i (l — e^
sin irp ( — 6j )

az

0 (1 — 0 — 1

4* • • •

A Cl

7T

?p(l-26!)

sin irp (1 — 2ex)

+

p~ (p + 1 — 2et) (1 — 2eJ

ira

0(1-260-1

sin 7r p (1 — 2ex)

+ • • •

- 2p(1"^ + S' {-^z(p, s ; ^ ’

sill 7rp(l - e2) s=_p \r bv o,2. . .

- z( 0 ; oz^p,

en e-2 • • \ ,

p '• h . . 1 + •••

By the employment of extended Riemann £ functions of parameter a, it is impossible
to give a form of this expansion which shall include all powers of a , analogous to the
expansion of log V (z + a), which involves Bernoullian functions of a as coefficients.
For brevity we content ourselves with the preceding first approximation.

§ 96. By differentiating the expansion for log P (2), given in § 68, we have at
once, as is evident by the preceding paragraph,

p/ ( s ) _ irp zP-i _ L

P (fi) sin 77-p ’ 2z

_ VM1 -O 2p (1-0-1

sin 77" p (1 — e1)

+ (-)'« F ( — - ) + ■ •

\ P

TTP~M1 ~ ^,(1-0-1

sin tt p (1 — e2)

+ (-)v-z(p,-p;.^;;_

Thus the asymptotic expansion for log P' (2) is given by

7 T

Sin 7Tp

irp\

+ [p — log 2 — log 277 + log — - + S' - — \ — F

V 2 / ® 2/d s & sm 77 p s=_„

sin 77/5 (1 — 6j)

*P(l-0

+

-P

pip + 1 ~2Pei)

S=-P

V -■

77

yP (1-20

sin 77/5 (1 — ea)

, 5/ l~)s 1 7 / . . 61> e2

+i, v z(^s’

ftp &2 • •

z(°; + terms involving positive

(fractional) powers of \/z.

d

Thus — P (z) is a function of the type

LIZ

CO

/ z \ - 2 + . .

(-) P*P1

n

n= 1

[{1+K)e

] 1

O

©1

where h„ — n1/p

b-. bn

1 + — + — + .
^ %£l T T

n

i/p

- - T- higher powers of —

/m 9 r

Thus the differential of an integral function of order p (> 1), where p is not
integral, is itself an integral function of order p whose nth zero, when n is large, will

ME. E. W. BARNES ON INTEGRAL FUNCTIONS.

499'

differ from the corresponding zero of the original function by the term - -y— y >

P

together with terms involving lower powers of n.

In an exactly similar manner it may be proved that the function R'p (2) admits
an asynrptotic expansion of which the dominant term is

(-)ppz'-2)0+p-3/2 exp. j(-)p-1 -p + log log z - i log 2tt + ( J8 F j ,

so that R'p (2) is of integral order p.

The term log log 2 in the exponential just written down shows that we shall come,
sooner or later, to a transcendental term in the expansion of the n,h zero of R'p (2).

Similarly the theorem may be established for the general simple non-repeated
function of finite integral order.

As regards the application of the same methods to simple repeated functions it is
only necessary to notice that corresponding to a zero k times repeated of the original
function there will be a zero (k — 1 ) times repeated of its derivative.

§ 97. We have now to consider whether the derivative of an integral function, all
of whose roots are real, can have zeros other than the real zeros which by the
extension of Rolle’s theorem separate the roots of the original function.

For this purpose let us consider the difference between the number of roots of P (2)
and of V (2) within a circle of very large radius r.

This number will be N = ~||y logP'fz) — ; log P(z)| dz

1 c d p' (z)

= - — j - log — . dz, where the integral is taken round the

circle in question.

Now by examining the various cases which can arise, it may at once be seen

d Y (z)

that the asymptotic expansion of y log " is given by (p — l)logz + terms which

_L l £ )

vanish when \z\ = x> . Therefore to a first approximation we have N = p — 1 .

If then the function P (2) is of genre p, its derivative can at most have only p zeros
besides those demanded by the extension of Rolle’s theorem.

And therefore when p is odd, P' (2) can at most have only (p — 1) imaginary roots.*
In particular when P (2) is of genre 0 or 1, P' (2) can have no imaginary roots.!

From this theorem coupled with the expansion given in § 5 and the equality

d J n (2) _ (2)

dz zn z'1

* Bokel, ‘ Fonctions Entieres,’ p. 44.
t Laguerre, ‘ G^uvres,’ t. 1, pp. 167 et seq.

500

MR. E. W. BARNES ON INTEGRAL FUNCTIONS.

j (z) . Jn (z)

we see that all the zeros of -±LrrL are real if the zeros of — — - are real — a theorem

/yll-T 1

<v A/

due to Macdonald.#

If the function P (2) be multiplied by an algebraic polynomial with real coefficients
whose degree is q, the derivative of the product can at most have only p-\-q imaginary
roots.

§ 98. So far we have only considered integral functions whose roots are all real and
negative. If, however, we have an integral function all of whose roots lie along
a line other than the negative half of the real axis, a change of the independent
variable will at once reduce it to an integral function all of whose roots are real and
negative.

If then an integral function of genre p have all its roots but q lying in a sequence
along a straight line through the origin, its derivative will at most have p + q roots
which do not lie along this line.

§ 99. We now conclude for the present the applications of the expansions which
have been obtained. There are many questions which are still to be discussed — for
instance : —

(1) Functions of infinite order ;

(2) Functions of multiple sequence ;

(3) Asymptotic expansions deducible from linear differential equations ;

(4) The rate of increase of the coefficients of the Taylor’s series expansion of an

integral function ; and so on.

Investigations in connection with each of these questions have been tentatively
undertaken — notably by Borel, Horn, Hadamard and Poincare. And I find it
possible to extend, by the methods of this memoir, many of the results which have
hitherto been obtained. But such investigations I leave for future publication.

* Macdonald, “Zeros of the Bessel Functions,” ‘ Proc. Bond. Math. Soc.,’ vol. 29, p. 575.

[ 501 ]

INDEX

TO THE

PHILOSOPHICAL TRANSACTIONS,

Series A, Vol. 199.

A.

Arc, electric, mechanism of (Ayrton), 299.

Ayrton (Mrs.). The Mechanism of the Electric Arc, 299.

B.

Barnes (E. W.). A Memoir on Integral Functions, 411.

Barnes (H. T.). On the Capacity for Heat of Water between the Freezing and Boiling-Points, together with a
Determination of the Mechanical Equivalent of Heat in Terms of the International Electrical Units, 149.

C.

Callenhar (H. L.). Continuous Electrical Calorimetry, 55.

Calorimetry, continuous electrical (Cal lend ar), 55; (Barnes), 149.

Cyanogenesis in Plants (Dttnstan and Henry), 399.

D.

Dttnstan (W. R-.) and Henry (T. A.). Cyanogenesis in Plants. — Part II. The Great Millet, Sorghum vulgare, 399.
Dynamics and statics, chemical, under the influence of light (Wildebjian), 337.

E.

Electric arc, mechanism of (Ayrton), 299.

Functions, integral, a memoir on (Barnes), 411.

F.

H.

Heat, determination of mechanical equivalent of, in terms of international electrical units (Barnes), 149.
Henry (T. A.). See Dttnstan and Henry.

Integral functions, a memoir on (Barnes), 411.

VOL. CXCIX. — A 320.

I.

3 T

18.11.02

502

INDEX.

J.

Jeans (J. H.). The Stability of a Spherical Nebula, 1.

L.

Light, chemical dynamics and statics under inliuence of (Wildebman), 337.

Nebula, stability of spherical (Jeans), 1.

N.

P.

Photo-chemical reactions, velocity of (Wildebman), 337.

R.

Reynolds (Osboene) and Smith (J. H.). On a Throw -testing Machine for Reversals of Mean Stress, 265.

S.

Smith (J. H.). See Reynolds and Smith.

Sorghum vulgare, glucoside and enzyme of (Dunstan and Henby), 399.
Stress, throw-testing machine for reversals of (Reynolds and Smith), 265.

T.

Throw-testing machine for reversals of stress (Reynolds and Smith), 265.

W.

Water, capacity for heat, between freezing and boiling-points (Babnes), 149.

Wildebman (Meyeb). On Chemical Dynamics and Statics under the Influence of Light, 337.

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