Shell revolving freely within a Hollow Spheroid. 427 



fluence of the force of attraction and its own centrifugal force, it 

 will under certain circumstances assume a position of equilibrium 

 in which its bounding surfaces will be two spheres concentric 

 with the spheroidal surface of the external solid shell. 



In order to show this, let the axis of rotation be the axis of z 

 of a system of rectangular coordinates, whose origin is in the 

 common centre of the spheres and spheroids. Let X, Y, Z be 

 the component parts parallel to the axes x, y, z of the attraction 

 which the hollow spheroid and the fluid shell exert on a particle 

 of the latter. The component parts of the attraction of the outer 

 spheroid may be denoted by 



— M#, —My, — ptJt, 

 and those of the inner spheroid by 



-Woe, -Mty, -Wz, 

 where M, N, M ; , and N' are quantities independent of x % y, z. 

 Let p' be the density of the fluid. Then the attraction which the 

 fluid shell exerts on the point x, y, z, has for components 



4 4 v v 



r 3 y 



-3* + f^7F* 



a_l_,,2_j_*9\8 



5# + 5 V/ 



3 fJ \/{a:* + y C2 + z*) s 



where r is the radius of the inner sphere. 



The component parts of the attraction of the hollow spheroid 

 and the fluid shell will consequently be 



liSlf^lp S^^^ t: (1) 



Z=(-^ + W^~7ro'f+~7rp ! f , — }z. 



V T 3 HJ ^3 r J t/( x 2 + y<2 + z ZfJ 



The differential equation for the surfaces de niveau, when w 

 denotes the angular velocity, is therefore 



(_ M+ M'-|v/+}v/ V p=qrpp+^)(^+^) 



2F2 



