Parameter Values of Potential Theory. 1 7.> 



It lias no piilf at /\= +1, while \= — 1 is not a singular value. 

 The series is therefore convergent for |A.1=1- In (36) we may put 

 \=±1 and thus obtain the densities of the simple strata which 

 satisfy respectively the boundary problems 



da 



The series for the solutions (21) may be obtained from that equa- 

 tion by substituting the values of PC'^p) and Q(pO). Further, if 

 Ao=I, the functions II+^{ts) and Cr-^^{ps) given by (20) have no 

 pole at \=1, while \=i — 1 is not a singular value. The series are 

 therefore convergent for jA| = l, so that the terms decrease indefi- 

 nitely. It follows that 



giving the electric distribution^. <j>r{t) in terms of the iterated 

 functions hn(ts) : the limit assuming one of m different values, 

 according to the surface upon which -'j lies: Similarly fiom the 

 convergence of the second series (20) for lA|=l, it follows that 



II = » 



i.e. 

 (37) V,.{t)=U g,,{ts) 



'll = CC 



giving the conductor potential Vr(fj as the limit of the sequence 

 gi{ts), g^i^-'^)- ■ ■ ■ which assumes m different values according to 

 the surface on which s lies. 



§8. — Solution of fhe second hnundary problem for both inner 

 and outer regions in terms of (t single function. — In the second 

 boundary proljlem the values A=±l correspond to the inner and 

 outer regions respectively. The former of these values is the only 

 pole involved. The boundary problem (19b) becomes, for Ao=U 

 and A=±l equivalent to the sepai-ate problems represented by 



^ ' ^ ^(t+)=~i(t) for A=+l 



dn 



'^^{r)=HO for A=:-l 



where in the former the boundary function i{t) is subject to the 

 usual integral condition. The solutions to the problems given by 

 (18) may be written 

 (39) r(p)=/Gll(pO)i(e)dO 



and 



v{p}=/vtl(pe)t(e)de 



1 Cf. Potentialtheoretische Untersuchuiiuen S. 5S>. 



