﻿GI Mr. H. Davies on the Solution of 



2. The proper solution for unbounded space is 



V=K„(mR), ....... (3) 



where 



E= V / r 2 + r' 2 -2r/ cos (0—0') 



and K„(#) is BessePs function of the second kind and of 

 order n. 



At this point it is necessary to introduce the relations 

 which hold between the various functions which will be 

 used. 



The BessePs functions of the second kind are related to 

 those of the first kind by the equation 



"When n is large the value of J n (#) is given very approxi- 

 mately by 



where II (n) represents an infinite product. 

 Taking the asymptotic value of II (n) we have 



J » (;C) = (1)" V2^!.U0 S ,.-,. - • • • <«> 



From (ft) it is evident that J n (x) vanishes at infinity 

 when the real part of n is positive. 



To obtain a value for J_ w (%) proceed as follows : — 



n(w)n(-n) 



sm^TT 

 sin nir 



II (—n) nir 

 Therefore 



n(«). 



J -" w= (l)"irF» 



.n 



«m nir 



This does not vanish at infinity when the real part of n is 

 positive. This difficulty can be overcome as follows. 



