MR. J. H. JEANS ON THE STABILITY OF A SPHERICAL NEBULA. 
Collecting results, and substituting for p x from equation (3), we find that equation 
(18) takes the form 
0A 3T, 
M r» st + ar 
i ra^ST , a/T, \ai’o 
+ ^o Vi l + K u ' -' 
dr dr 
0 dr \2Tj dr 
0T n /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 = It 0 , 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 T 1; starting from any arbitrary values of u, A, T x , 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'T 1 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 
