202 



Prof. A. Gray on Electric 



breadth dp (fig. 1). Let; E be a short element of this ring, 

 and <j) be the angle which the radius OE makes with the 



Fier. "!• 



/ 



\ 



/ 

 / 



\ 



\ 



b 





"\c \ 





/ 



\ 

 \ 



/ 

 / 

 / 



radius which lies in the plane PCO. The potential V at P 

 is given by 



v = <T rr ^^ . m 



Jo Jo ^+y 2 +P 2 -2 P ^co S 0)* v ' 



Put R 2 = y 2 + p 2 — 2>jp cos <p. We get (see Gray and 

 Mathews, ' Bessel Functions,' p. 72) 



V=a\ d<t>\ pdp\ e-^J (\R)d\. ... (2) 

 Jo Jo Jo 



But by Neumann's theorem (G. & M. p. 27) 



J (\R) = J (\p) J (\v) + 2S J (\p) J s (\y) cos scf>. (3) 



Hence, since the order or! integration in (2) maybe changed, 

 we obtain 



J (\R)flty = 27rJ (\p)J (ty), ... (4) 



since the terms involving cos scf) contribute nothing to the 

 integral. Thus (2) becomes 



Y = 27r*\ e-**f( pdpJ QLp) \j (\y)d\. . (5) 



Now it is easy to prove, by the relation 



sJi'(*j+Ji(*) = «Jo(*), 



or otherwise, that 



fa 

 pdpJ (\p) =~J 1 {\a). 

 « o ^ 



