36 Proceedings of the Royal Irish Acadenuj. 



The equations asserting equivalence of the potential and its differential 

 coefficient to r, for r = give now 



B„ = - nRY,{nR) {P„ + -^sin ^' = I - (35') 



whence 



V=a^- 4X,K,{nR) Y,{nR) sin^ - 2X,P,,K,{nR) Y,{nR) sin nl 



(36') 



In evaluating now P„, by substituting as before in it, for £,^', we may 

 distinguish two cases : 



I. tnot<R. 

 Here, by the substitution of we shall have 



Q = 2,.'-^^, K,in'R)Y,{n'R). 



where 



"We may evidently, since / becomes indefinitely large, write / for 

 I- ^; so we shall find 



_ 16aR- sin 71^^ am ^ 



= f sin^- Y,{x')dx' ; 



%P^K^{nR) Y,[nR) sin = — ^ %,,a„,QJ. 



If, now, in the integral we write for sin-— its mean 

 value = ^ q.p., we shall have 



r= a^-42„ir,(»iE)r,(»ie)^^sin=(»c-8^,2„a.„ jj" '^^ ^a^Iff j" 



ii'l 127r 

 iN'ow it is easily seen that 



rr 



