﻿and on the Gravity-Potential of a Ring. 



471 



If the density of the sphere is variable, the only difficulty 

 is that of finding the particular integral /. As an example 

 of variable density, take the case of a sphere in which the 

 density varies as the square of the distance from a diametral 

 plane. Here 



_V 2 4>=-47rc; 2 , 



/= — 6 TTCZ* = — K7TCr 4 COS 4 0, 



o 6 



and the general solution involves zonal harmonics (P»). 

 Hence, inside, 



$ = - 1 ttcv* cos 4 0+2 B n r« P n , 



and outside, 



r n + l 



Now, a 8 D 4 D 1 



The surface conditions give at once 



A n =0 = B n for all n > 4, 

 and A 1 = A 3 = = B 1 = B 3 . 



Also 1 4 -p A 



— r^ TTCa + B = 



15 a 



- A Trca 4 + B 2 ci 2 = ^| ^ ^ = </>' when r = a, 

 L l a 



and 



4 o A 



15 a 2 



16 



"Si* 



,A 4 



-iQ5^a 3 + 4B 4 a 3 = 



giving 



* /wc = 15 r~ + 105 ? Ps ; 



<f>'/7rc = l(5a 4 -r 4 )+ i^(7aV-5r 4 )P 2 . 

 10. The gravitation potential of a tore has been recently 



