*• 
STABILITY OF A GEAVITATING PLANET. 
while the value of ih and may be written in the form 
Vi 
+ 20/d + 21n + 7 
(2?i + 1) (2a + 2) (2a + 3) 
2 («, - 1) ( 2 a + T)^ 
L[X 
■ (59). 
pft” a 
^ 7” + 1) + 3) 
(GO). 
The equation giving points of bifurcation can now be found by substituting these 
values in equation (54). 
§ 17. This equation will have roots corresponding to the different integral values ol 
n, n = 0, I, 2 . . . ; these determine points of bifurcation such that the critical 
vibrations are of orders = 0, 1, 2 . . . respectively. 
Of these the points of bifurcation of zero order are of no importance. The reason 
is exactly similar to that given in the case of a splierical nebula (§28 of the paper 
already quoted),* namely, that a point of bifurcation of order n = 0 does not indicate 
a departure from the spherical shape. We therefore will only discuss values of n 
different from zero. 
Case uf y 0. 
§ 18. Before discussing the general 
to consider the simple case of y = 0. 
and (58) 
form assumed by equation (54), it will be well 
Putting /X = 0, we obtain from ecpuitions (57) 
\ fji. 
H+h {^)- 
Pteferring to equations (52) and (53) v'e see that the equation givnig jx.tiiits of 
bifurcation is 
= «.(« 1 )- 
The hjw'est roots of \ arious (orders other than zer(j are 
'l l 1, 2, 3, 4, 5, 
a; = 4‘4'J, 57G, G'98, 8-18, 9-37, &c., 
the roots continually increasing Avith ii. Thus the first point of bifurcation is given 
by X = 4‘49, and the critical vibration is of order n = 1. 
Case of IX Different from Zero. 
§ 19. The general equation in which p, is not put equal to zero is much more 
complicated than equation (61), which has just been considered. If we write ii,t for 
J„ + i (^)/J«-4 (^)’ equation giving points of bifurcation of order 
n is of the form 
= an algebraic function of x and of (X -f- 
VOL. OCX.—A. 
Z 
