176 G. E. Weatlierhurn : 



respectively, where the index represents the pole Xq= + 1 and the 

 suffix the particular value of A. As noAv the pole Xo= + 1 is the 

 •only one to be considered we may drop the index in what follows. 

 These two solutions are expressed in terms of different functions 

 <T + \{ps) and G^iips). It is our object to express both of these in 

 terms of a single function. By means of the second equation (8) 

 w^e may write 



(40) ( G^,{ps)^/G^,{pe)h{6s)d6=g{j^s)-T{r) 



» G-,{vs)+/G^,{pe)h{e.^)de^g{ps) - v{p) 



If we put 



{1B{ps) = G^r{ps) + G i(;«) 



\ 2R,{ps) = G^,{ps)-G ^,{ps) 

 we obtain from the preceding by adding and subtracting 



(41 ) ( R{ps)-fR,Xpe)h{es)de=g{ps) - Tip) 

 ' Ri{p^^)-/R{pO)h{0s)de=O 



This last equation expresses B^ps) in terms of B(ps}; hence we 

 may determine both G + i(ps)iind G-.\{p^) in terms of the single 

 function S(ps). From (41) we find rliat F{p.<i) satisfies the integral 

 equation 



B(ps) -/R(pe)h,{Os)dd=<j(ps) - T{p). 



As in §4, by the method of successive approximations, this integral 

 •equation gives us an expansion for R{ps) and hence for R^(ps). We 

 find 



( R{ps) = lffip.)-V{p)] + \!UP'^)-r{p)] + [ff,{ps)-T(p}]+ . . 

 \ Ii,{P^^)=[y,{px)-V{p)]-\-[g.,lps)-T(p)]+ . . . 

 Avhich are both convergent, being identical with tliose obtained by 

 adding and subtracting the absolutely convergent sei'ies for G + i{ps) 

 «-nd G- lips). 



The solutions of the second boundary problem for both the inner 

 and the outer regions could also be expressed in terms of the 

 function K(ts) introduced by Plemelj.i For from (8) we find 



G + \ps) - \l\j{pe)U^^ds)d6=z<i{ps)-T{p) 

 In this we may putAr:r±l in turn, and thus obtain G + iips) 

 and G-iips) in terms of H+i{ts) and 7/_i(te) respectively, and 

 hence in terms of li(fn). Introducing the values of tlie functions 

 we find 



G + i{ps)=y{ps)- Tip) +/;/(]>e)\ K(Os) +/h{6(r)K{crs)dcT\de 

 = yips)-Tip) +/g(pe)K(ds)d6 +/g,(pO)K{Os)dO 

 Similarlv 



G_,ips)=;,{ps)--Tip) -/(/ip6)Ki0s)d0+/;/,{pe)Ki0s)d6 



So that the solutions for both legions may be expressed in terms 

 of K(ts). 



1. Potent. Unter. S. 79. 



