STABILITY OF A GRAVITATING PLANET. 
163 
We shall assume this provisionally, in order to avoid the continual repetition of 
double summation, and now proceed to evaluate rj, ^ and to form the boundary 
equations for the simple vibration given by equation (13). 
§ 10. From equation (8) it appears that the displacement f is given by 
^ ^ af“. 
The solution is 
+ .( 1 ^)’ 
where is any solution of 
p^p(f> + = — (A. + p.) A 4* pE.(li")) 
and is the most general solution of 
+ ,xV’f„ = 0.(18), 
It can easily he verified tliat a solution of equation (17) is 
.•> 
jr 
\ + 2 p 
P 
A 
( 1 !)). 
There will he solutions for rj, (, similar to (16), hut the three solutions for rj, 'Q 
must he such that 
y + ';’' + 'f=A.(20). 
ih'. cljj dz 
The left-hand member of (20) is, from (16), 
+ f + 
1 - 1^1 , 
dy dz 
and from (lO) and (17), tlence (20) is satisfied if 
I I 
^d.c dy 
dz 
- 0 . 
( 21 ). 
§ 11. Write u for — £+ -f- as before, and ((q for -f '^ 
we shall verify that the solutions for u and u^^ are 
u — aS„, 
((q - 
in which «, are functions of r, as yet unknown. 
Assuming these solutions, the value of E, calculated as explained in § 6, is 
•pi _ "iTrpS^j J 1 
2 » + 1 [r 
J«+i {xr) dr -f r (/cr) dr + ^ 
Then 
( 22 ), 
(23), 
Y 2 
where denotes («),.=„. 
