﻿472 Self-induction and Gravity -Potential of a Ring. 



considered by Mr. Dyson * by means of special functions and 

 integration. It may be of use to indicate how the problem 

 would be solved by toroidal functions and the differentiation 

 method illustrated in this paper. 



Inside the tore — V 2 </>= — 47nr with a constant. A 

 particular solution is 



/= -7TC7/) 2 = -TTO-^S^C-C)- 2 . 



Hence : — 

 Outside, $ = s/(C—c)%A n ~P n cosnv, 

 Inside, ft = -7r<ja 2 S 2 /(C-c) 2 + V (C-c)lB n Q n cosnv. 



f must be expanded in a series of the form 



/= N /(0-c)SF B coswu; 



whence 



Jcosm; 



This has been found above (9), viz. 



F n =-^.^(SQ',!-CQ'„), 



F =-i^.^(SQS-CQ' ). 



Writing P n for R„ and Q„ for T n in § 3, we find 



a„(PLQ,-p„Q'J = QA-Q'A| 



But p:.q»-p„q;, = I- 



Hence A. n = ^(Q.F-Q'.F,), 



IT 



B = -(P F-P'F ). 



n \ n n n nl 



7T 



|F n =-^a 2 



3 



Inserting the values of F n and F^, we get 



°{( n2 -i) Q "-w- Q » Q »- Q »} 



A.-i^l™-"* l\„ C* + 1 



B>^^0{(-'-J)S- + (-.-J)P A -^P A - I iQL}. 



The external and internal potentials are thus completely 

 found. 



* " The Potential of an Anchor Ring," Phil Trans. 1893, pp. 43, 1041. 



