32 Proceedings of the Royal Irish Academij. 



Hence we have 



3 J„ x' 



... J'^. i-^^)^W .4. 



Again, equating the value of the potential given above for 2 = 

 with its value previously found, we have 



TT o 1/ lb 



Proceeding as before, we have then 

 or 



an expression connecting the second elliptic complete function E with 

 Bessel's functions. 



If, in this formula, we put r = 0, we have 



if, again, r - 



(fir 

 (i-^F,wir.(^))-. (28) 

 * 



"We have also 



= - ^ Rd%K,{nr) Y^nR) = _ . | ^ Y,{x) dx. 



But also 



dr TT A \Rj \Rjjr 



whence 



'rx\ R( r\ ^( 



(29) 



