152 Compressible and Non- Homogeneous Masses [CH. vn 



For a spherical configuration (o> = 0) there is obviously no limit to the 

 value of V A : the critical value v A r is infinite. But as o> increases V A ' will 

 diminish, and matter will be ejected from the equator as soon as V A = V A '. 

 Let us examine the value of the critical value V A . 



Let the' spheroidal or ellipsoidal figure of the nucleus be supposed to be 

 the standard ellipsoid 



the axis of z being the axis of rotation. At a point x on the prolongation 

 of the major-axis, the gravitational attraction is 



f 00 d\ 



A = ATTpQabcx I ^ Y r. 



> * 2 -a2 (a 2 + X)* (6 2 + X)* (c 2 + X)* 



For a Maclaurin spheroid, in which a = b, the integration can be effected, 

 and we find 



[1 a tg g2\&~i 

 sin" 1 / , (403) 

 ft X GL* ' X 



where a = (a 2 c 2 )*. The ratio of centrifugal force to gravity at any point 

 on the #-axis is tfx/X. At a point on the boundary of the nucleus, this ratio 

 is always less than unity, but it increases as we pass outwards, and the point 

 at which it attains the value unity is the critical point at which 9fl/8 = 0. 

 Hence to obtain this critical point, we must equate the right-hand member 

 of equation (403) to wrx ; the resulting equation is 



ft) 2 fl a Oz 2 -a 2 )l 



-=a6c -sm- 1 --^ 

 2?r/3 |_a 3 x tfx z J 



(404). 



The value of x which satisfies this equation determines the radius of the 

 equator of the limiting equipotential. 



In the special case in which the nucleus is a Maclaurin spheroid at its 

 ellipsoidal point, of bifurcation, the value of ft) 2 /27r^ is 0'18712, and the root 

 of equation (404) is found to be 



x = 1-6436 a = 1-5990 (abc)*. 



The critical equipotential is drawn in fig. 29 ; it is clear that the value of 

 V A here is quite small, being in point of fact rather less than one-third of v y . 



Fig. 29. 



