﻿and on the Gravity-Potential of a Ring. 461 



therefore 



Ftt = ^p^ {(QL-QL.)-(QL« -Q«)} 



= 87r6aV2f(n-l)Q n _ 1 -(n + l)Qn+i}; 

 also 



F =-8tt^V2Q 1 . 



It is now only necessary to substitute these values in 

 equations (3). Then 



+ (^-i)SQ n {(n~l)Q n _ 1 -(n + l)Q n+1 }]; 



since 



T.= -S% and T , .= -(n«-i)8Q»- 



Now 



(n 2 -i)Q„Q B _,-Q'„QL> 



= ^±iQ n {4nOQ,,-(2n+l)Q. +1 }-QL(<X + .-2nSQ,)j 



since 



(2»+l)Q„ +1 -4«CQ„ + (2n-l)Q„_ 1 =0| 



Qn+i — Q»-i = 2nSQ„. J 



Therefore 



(» 2 -£)q»q»-.-q'»ql, 



= - (2 "^ 1)3 Q,»Q a+ ,-Q , „Qn + .+»Q„{(2n+l)CQ„+2SQU 

 = {»(2„+1)-(^±11- 2 }q„q, i+1 _q^ +i 



=( w2 -j)Q»Qn +1 -Q'„Q'„ +1 ; 



therefore 



TX-T,F.= - 16rfaV 2 { (» 2 - j) Q„Q„ +1 - Q'„QU. } ; 



whence 



166aV 



A n =- 



n 2 



X~ { ( ^ ~ j)Q»Q»+ 1 — Q'nQU 1 J 



= 16&aV2a„ (say) 

 with A =-86aV2(QoQ 1 H-4Q' Q , 1 ). J 



(6) 



