Pantiiietrr Vulues of Potential Theory. 171 



and the tirst c(|uati()n ("iO) becomes 



("^') |^(^y;)=.[/K//>)-|T(<;.)]+A[A,(/;>)-lp(/p)]+ ... 



Introducinp' these values in (18) we have, for the solutions of the 

 boundary problems (li)) 



(21) I w{p)=/no){[h{ep)^lF{ep)'] +x[a,(^p)-1p(p^)] + .. }d^. 



) np)==/{[y{j>0)-Q{pe)] + x[g,ip6)-lQ{j>0)'^ +..]HO)dO. 



We may further obtain expansions for the moment i'(/,), and the 

 density [ji{t) of the strata satisfying (19); for these are solutions of 

 the integral equations 



( v{t) - Xfv{6)h{$t)de=i{t) -/i{0)F(Ot)dO=^{t), say 

 I ix{t) - X/h{tO)fjL(d)dd=i{t) -/P{tO) iiO)dd = F{t), say, 



and are therefore given by the expansions 



(22) I v(0=E(0+AEi(0 + A%(0+ .... 



) fM{i)=F{t)+\F,'{t)+X,F,'{t)+ .... 



where the successive functions are given by 



E,{t)=/F40)h(ef)dH 



E.,(t)=rE,{6)h{6t.)d6, &c. 

 and 



¥,'(()=/ h{te)F{6)de 



F.J{f)=/h{td)Fi'{0)de, &c. 

 If we evaluate these functions we find 



Ea{t)=/l(e)h„,_-^(6t)dO-\/i{0)F{ef.)dO 



Fn'(0 =J'K- imi {0)de - ^/P{fO) HO)dB 



If now we form double and simple strata with moment and density 

 given by (22) we find exactly the series (21) over again. 



§5. — Forimd(E of Reciprocity. — The Green's function G(p5) 

 admits certain theorems of reciprocity. The argument used to 

 establish these for the ordinary potential^ is equally valid for the 

 generalised, the symbols having their altered significance. These 

 relations may be stated 



i. If the points p and q are both in the same region or both on 

 the Ixiuiidary 



(23) G(/^) = C(y/) 



1. Pleraelj. F.oc. fit., S. 39fi-3i)8. 



