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[ ]

CONTENTS.

(A)

VOL. 199.

Advertisement

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 of the International Electrical Units. — Experiments by the Continuous- Flow 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, D.Sc. Lo7id . 399

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., LL.D., F.R.S. . 411

hidex to Volume

50 1

[ V ]

A D TERTIS E M E N T.

The Committee appointed by the Royal Society to direct the publication of the Philosophical Transactions take this opportunity to acquaint the public that it fully appears, as well from the Council-books and Journals of the Society as from repeated declarations which have been made in several former Transactions, that the printing of them was always, from time to time, the single act of the respective Secretaries till the Forty-seventh Volume ; the Society, as a Body, never interesting themselves any further in their publication than by occasionally recommending the revival of them to some of their Secretaries, when, from the particular circumstances of their affairs, the Transactions had happened for any length of time to be intermitted. And this seems principally to have been done with a view to satisfy the public that their usual meetings were then continued, for the improvement of knowledge and benefit of mankind : the great ends of their first institution by the Royal Charters, and which they have ever since steadily pursued.

But the Society being of late years greatly enlarged, and their communications more numerous, it was thought advisable that a Committee of their members should be appointed to reconsider the papers read before them, and select out of them such as they should judge most proper for publication in the future Transactions ; which was accordingly done upon the 26th of March, 1752. And the grounds of their choice are, and will continue to be, the importance and singularity of the subjects, or the advantageous manner of treating them ; without pretending to answer for the certainty of the facts, or propriety of the reasonings contained in the several papers so published, which must still rest on the credit or judgment of their respective authors.

It is likewise necessary on this occasion to remark, that it is an established rule of the Society, to which they will always adhere, never to give their opinion, as a Body,

[ vi ]

upon any subject, either of Nature or Art, that comes before them. And therefore the thanks, which are frequently proposed from the Chair, to be given to the authors of such papers as are read at their accustomed meetings, or to the persons through whose hands they received them, are to be considered in no other light than as a matter of civility, in return for the respect shown to the Society by those communications. The like also is to be said with regard to the several projects, inventions, and curiosities of various kinds, which are often exhibited to the Society ; the authors whereof, or those who exhibit them, frequently take the liberty to report, and even to certify in the public newspapers, that they have met with the highest applause and approbation. And therefore it is hoped that no regard will hereafter be paid to such reports and public notices ; which in some instances have been too lightly credited, to the 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