STABILITY OF A GRAVITATING TLANET, 
IG 5 
xS„ 
r 
2 «T 1 
d 
diz 
1 
‘In + 1 
d 
Jr 
■/) 
0) 
From these identities it is clear that if the terms in (27) which do not depend on 
or Uq are expanded in spherical harmonics, they will contain no harmonics other 
than oj and y. We therefore see that the form of rnay be assumed to be 
= + .(^ 8 ), 
where and arc functions of r. The value of Uq is 
= 1)©)S,.(29), 
whence 
= 1)<Q.(30). 
§ 14. Substituting for in (27) and eipiating the coefficients of w and y, we obtain 
the two following e(piations which must lie satisfied at r = a :— 
V ?'VT„ + J (Ar) 1 
2 /i + 1 
/X 2/t T I 
d 
j,— («+ 0 / y1l + 2 
dr 
dr 
d 
d 
+ 
47r/5(2a - 2) fr' 
(2a + l)ffi Vy 
-i-i 
r , 4 - — ^_ ,.-0 + i ) T /,.»+2 
““ + 2a + 1 ' 
d 
J 
'■.^('•”A) + r'')“- 5 a = o 
(31). 
and a second equation of a similar kind, of which the first line can he obtained from 
the first line of the above by writing — [n -f 1) for n, and the second line is 
I / n \ 
■J.,1 + 1 (/)■ ' ' ' d, 
llie expression winch xiccurs in cui'led Ijrackets in (31) can lie transf irmed into 
d 
"’v .ffl)i 
while the correspxmding expression in (32) is seen to he 
d i \ d 
+1 
(33), 
from ihe value of (fi, given by eipiation (24), 
d 
dr 
(..i.qjT) ^ Cb/ '-ffi,_.(«r) + { 2,1 + l) L)rM,_,(/<o), 
£(r-”^)= - CKr-^-^)J,„jKr). 
