﻿and on the Gravity- Potential of a Ring. 465 



l()7r^W2* 2 f /- , 7 M , 81 \ „ 



+ ^-k 4 R 2 cos2v^ . .j 



8ttRIv^_cos2« l_ (o\ 



z l + cos« */(C-<Ol J' 

 The total flux through the aperture is the value of ty at 

 the inside edge, i.e. at u=u 0i v — ir. 

 Then 



I^Mco^a ^r / 7 g y X B 

 r 1 + cos a 1 + A; (. \ 4 32 / 



~H 1+ ir* s ) H » 



Substituting for the R it will be found that 

 , _ 8ttRcos 2 « r T a 3 7 1/ T 1\ 7Q 3/ 0T 3l\ w 1 



which for very small rings gives the self-induction 



= 4ttR(L-2), 



the value usually accepted for the true value. 



The flux as taken by Prof. Minchin is found by putting 

 ^ = 0, when 



, /T 8ttRcos 2 « r T a 3 7 n / T 23\ 79 3/ OT 31\ ; ,1 



This agrees with his result up to the term in k. 

 The force at the centre of the ring is 



1 df du a a 



s — y-.-r- when u = and v = ir. 

 zirp du an 



This 



= 2^(V/ [" 2(&) + V(t) 2 ( n8 -D A - SP » C ° S Wt? ] 



= 326 { \ (Q^ + 4q;q;) + ^(-lj^QLQL+i-S^lQ^Q^i)} • 



