﻿and on the Gravity- Potential of a Ring. 463 



therefore 

 E = - 1 tt 3 6 2 R 2 (1 - cos a)*(l + 2 cos «) + 2 V 2abXB n i ~ {Q! n )*du. 



The integral may be evaluated as follows : 



Q' n+1 -Qn-i = 2nSQ n . 

 Also it may easily be proved that 



Q , n + i + Qn-i = 2CQ' n +SQ, l ; 

 Therefore 



(Q'„-h) 2 -(Q'»-.) 2 = 2nS(2CQ„Q'»+ SQ^) 



therefore °^ u 



£ |(Q'„ +1 ) 2 A«- f g(QL,)^«=[2nCQ 2 j; 



or, say 



= -2nCQ;, 



X, l+ i — X 7l _ 1 = — 2nCQ n ; 

 therefore 



X n = 2C{(n+l)Q^ +1 +(n + 3)Q 7 2 l+3 + ...K 



an infinite and rapidly convergent series. 



Substituting this and the value of b from (5), 



^ 7rRI2 M o > , R ^ 2 Pcosa v B n X n 



-^ = a~ (1 + 2 COS a) + — ^ j- rg X 2Tj2A , 



6 ' 2 (1— cosa) 2 7r 2 it^6 7 



and the self-induction, 



^R/i , o \ . oa t> COS 3 a ^ /3 n X n 



= -^-(l + 2cos tt )+32^R. (1 _ cos ^ S-^. . 



Now P„ = «„E' + /3„F', 



Q„ = «„(F-E) + /3„F, 

 where E, F are elliptic integrals to modulus k ; where 



1 — cos a 



2 _ 



1-fCOSa' 



E'j F' are the complementary integrals. 



Also a n , /3 n are algebraical expressions in k whose values 

 are given up to n=5 in |_T. F. ii. 8]. Their general values 

 are also given in a paper by Mr. Basset *. Consequently the 



* "On Toroidal Functions," Amer. Journ. Math. xv. p. 301. 



