672 Dr. J. R. Wilton on Figures of 



Excluding the case where the surface extends to infinity, 

 we may change the order of integration in this result, and 

 obtain, after a slight reduction, 



V = 2p\ a ^i 7r ^\ v /^ 2 +(e-?+^KcosX) 2 -27rp(a 2 + c 2 ), 



J— a Jo 



where it is assumed that the origin is the middle point of 

 the axis, and that <st = when f= ±cr. 



The equation which has to be satisfied on the surface, 

 namely 



V 4- -|&) 2 R 2 = constant, 



&> being the angular velocity, is thus 



17 J -a *JQ P 



or in spherical polar coordinates, 



- f da ( ^ d\~\rx) sjr\\-x 2 ) + (z + iR cos X-raf 



where r is the distance from the origin of a point on the 

 surface, and a is the cosine of the angle which r makes with 

 the axis. 



If we put £ + *Rcos\ = £, we have | f | = \/.s 2 + R 2 cos 2 A > 

 which is less than \/z 2 + R 2 , i. e. less than r, since z, R is a 

 point on the surface. Hence we may expand the left-hand 

 side in powers of f/r. We have then, putting 



1 h <°* \ 



C + vr 2 + (l-p)r 2 FJa)=:- \da\dxA (rx)\/r^M^+l 2 



tt J _ J da ' 



where P n denotes the wth zonal harmonic, the argument 

 being a. 



