STABILITY OF A GRAVITATING PLANET. 
1157 
15. Now is known to be a solution of equation (18), and hence, from equation 
(28), we may assume 
(hr), © = {hr), 
h^=]^p/l^ . 
It follows that at r = a we may write 
where 
(42). 
wliere 
J 
« - = B®, 
A = a -- log {<1 (ha)) . 
B = a log (crhJ„+3 {ha)) . 
Hence, from (30), we have at r = «, 
da., 
a 
dr 
" = r,Ap - (n + 1) B® 
(43) . 
(44) . 
(45) . 
Lastly, we have from (28) 
L = 1 I>-^"''|,(>-"S.) + Q>-" ’£('■-'""” 8 .). 
In order that (21) may he satisfied, we must have 
^(P,-’*+-) (n + 1 ).= 0 
dr H/' 
(46). 
Substituting for and (1^, we find that this is satisfied for all values of r it 
«fo+(” + 1)®0 = «.(40. 
® - ep.(48), 
.(43). 
fie., if we have, at r = o, 
where 
^ _ r._ J„ + 3 {hn) 
V + 1 'I«_i {Iifi) 
Equations (40) and (41) now assume the forms 
^+ 4vp(2n-2)r ^ 
p~ fir 
4:71 p (2n — 2) r 
_g 1) + (^3 _ 1) + (,,3 q_ 3„ q_ 2) 
+ (3)? + l) A-(n+ l)Bd 
= 0 . 
. (50), 
F + P[u —«^ + (w2+2n)6' + ??.A —(3/? + 2)B(9] = 0 . . . (51), 
in which r must be put equal to a. 
The general frequency equation may be found at once by tlie elimination ot p, the 
values of and B being given by equations (36) and (37). 
