426 Mr Basset, On finding the Potentials of [May 10, 



2. The first step is to establish a theorem analogous to Fourier's 

 theorem, for expressing any function in the form of a definite 

 integral involving Bessel's functions. The theorem is as follows : 



If p and q be any positive real quantities, and cp (p) is a function 

 which is finite and continuous for all values of p which lie between 

 the limits p and q, but which is not necessarily finite at the limits, 

 then the definite integral 



r°° rp 



I dX I Xu<p (u) J m (\u) J m (Xp) du 

 Jo Jq 



is equal to cp (p) when p lies between the limits p and q, and is equal 

 to zero when p lies beyond these limits *. 



In order to prove the theorem, consider a thin plane conductor 

 bounded by two concentric circles of radii p and q, which is 

 electrified in such a manner that the density on either side is 

 equal to 



\ cp (p) cos mcp. 



The potential will be 



V 



J a 



p r2Tr+<}, U( p ( u ) cos rnc\> dudcp' 



[z 2 + p 2 + u 2 - 2pu cos (cp' - cp)} 2 



Let 4 > ~ = V 



R 2 = p 2 + u 2 — 2pu cos 7). 



m . T ^ [p /" 27r ud) (u) (cos nicb cos mn — sin mc\> sin mrj) dudv 

 lhen V = — — - — ; • 



JqJO (Z 2 + R 2 f 



The second integral vanishes, and the first is equal to 



2 cos mcp I dX I dul e -As utp (u) cos mnJ (XR) dn -f\ 

 J o J q J o 



Now J (XR) = J (Xp) J (Xu) + 22" J n (Xp) J m (Xu) cos mn, 



v 



whence V = 2ir cos mcp \ dX I e~ Xs ucp (u) J m (Xu) J m (Xp) du. 

 Jo J a 



The density ~~1l~ ~J~ > 



hence this quantity must be equal to \c\> (p) cos mcp when p> p>q, 

 and must be zero when p lies beyond the limits p and q, whence 



! dX\Xucp(u)J m (^u)J m (Xp)du = cp(p)p>p>q (6) 



Jo J q 



= P > H (6a). 



or p < q) 



* The particular case of n—0, 2 = 0, p-<x> is given in Heine, Kugelfunctionen, 

 Vol. ii., p. 299. 



t Since / V Az J (Xtf ) d\ = (* 2 + i? 2 )-* . 



