﻿and on the Gravity- Potential of a Ring. 459 



yjr must be finite at all points inside the ring, and must satisfy 



d*+ dfy ld+ _ , ... 



-dT* + d?~~pdi = ~ S ^ ■ • ' () 

 These conditions are satisfied by 



where/is any particular solution of the equation (1). 



Further, at the boundary of the ring yjr and the force are 

 continuous. That is, when u = u then for all values of v, 



♦-* - Sf-5& • • • • « 



where w is the value of u corresponding to the boundary. 

 These conditions will serve to find completely yjr and yjr' in 

 all cases. 



In order to apply these last conditions it will be necessary 

 to expand / in a series of cosines and sines of multiple angles. 

 In the cases here considered it will be seen later that the sine 

 terms do not enter, and that /is of the form 



/= ^(C- C ) SF » C0SH "' 



where F n is a function of u only. 

 The conditions (2) now give 



A f n = 0, B' n =0. 

 A B B*=B n T n +F„, 



A n R„ =B n T„ + F„; 



where dashed letters denote differentiation with respect to u, 

 and the functions are now to be regarded as functions of ?/„ 

 only. 



From these, 



A„(R„T„ — R„ r F n ) = T n F n — T„F„, 



B // (R / w T w -R H T , n ) = R n F' w -RU^. 

 Now 



R'„T n -R n T' M = (^-I)S 2 (P f , 1 Q K -P„Q' rt ) [T. F. ii. 2.] 



= 2 -j)7rS. [T.F. i. 24.] 



2 12 



