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 (11), 
(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 
Ap 0 A u 
d Po \ 
dr ) 
dPo 
dr 
-fi Bp 0 ) 
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 = 
47r 
(in + 1) 
^ {- j k (A % + Bp )dr - [A(p - <7 0 ) 
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 l y .(28). 
v ' p dr 
Substituting the assumed solutions for u, A and If, and the corresponding values 
for y, p l5 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. 
