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 V 0 + V' and V 0 , 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 = R 0 , the surface 
density being 
- [■ u( Po - o- 0 )] r = Ko , 
where <x 0 is the mean density of the core, and 
(ii.) The potential of a surface-distribution over the sphere r = R l5 the surface 
density being 
[«(P0 ~ 0‘l)]r = B 1 , 
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 
(V 0 + W) - 
1 0 
(Po + P) 
+ cfi). 
(7). 
Transforming to polar co-ordinates, these equations are equivalent to 
9% 
ot“ 
|(V„ + V')- 
(po + p') 
( CT 0 + *0 • 
( 8 ). 
0% _ 1 0V' 1_ dm' 
dt 2 r d9 p 0 r d6 
(9) 
. . dho 1 0V' 1 dm' 
r Sill U -TTT = 7—7. w-;— - 
ct~ r sm 6 c</> p 0 r sm 6 cep 
As an equation of equilibrium, we have 
<Wp _ 1 0OTq _ 
dr p 0 dr ~~ 
( 10 ). 
( 11 ). 
and with the help of this, equation (8) reduces to 
dru 0V' 1 dm' p' 0w o 
dt~ dr p 0 dr p 0 ~ dr ‘ 
as far as the first order of small quantities. 
* Boltzmann, ‘ Vorlesungen liber Gastheorie,’ vol. 1, § 13. 
