Parameter Values of Potential Theory. 167 



If again we substitute from (6) in (5) and use tlio i-elations (7) 

 we find — 



^^^ J xfh{td)H{ep}d6=x/ii{te)h{ep)do=ii{tp)-.h{tp) + ~ .p(^;>) 



I \fg{qd)H{dp)d e= \rG{qe) h{dp)d 6= G{qp) - cj{qp) V Q.{qp) 

 These relations are more general than those found for the ordinary 

 potential by Plemelj, who considers mainly the pole A= +1. They 

 play an important part in our argument. 



The value of 'P{ts) is known, being the residue of tlie resolvent 

 for the simple pole Xq- I^ '"■ ^^^ the order of multiplicity of the 

 root Ao of the determinant D{/\), P(^s) may be expressed as the 

 sum 



(9) P{te) = <^,(0<Al(«) + Ut)Hs) + • . • + <t>m{t)^m{s) 



where the functions ^i, i//i(i=:l, 2, . . . . m) are the m linearly 

 independent solutions of the homogeneous integral equations. 



<ji{t)=Xo/h{te)<f>{e)de 



x(f{t) = ko/^{0)h{dt)de 

 satisfying the usual orthogonal relations. Hence the values of P{tp) 

 .and Q,{qp) are given by 



(9') j -P{tp) =cfy,{t)4,,{p)+ +<f>mmm{p) 



\ Q{qp) = ^i{q)hip)+ +^m{?)MP) 



where ^(q) is the potential of a simple stratum of density (j>{t) over the 

 boundary, and i/^Qj) is that of a double stratum of moment Xa^(t). 

 If we introduce the functions 



^^^) jk{tp) = h{tp)-\-P{tp) 



we are enabled to express (8) in a form exactly similar to (5). For 

 if in the first of (8) we replace p by 0, multiply throughout by 

 V(Op) and integrate over the boundary, we find in virtue of (T) 

 that 



/H(te)V{ep)dd =/F{td)H{Op)de = o. 



Similarly it may be proved that 



J G{qd)V{dp)dd =/Q{qe)H{ep)d6 = 0. 



These integrals may therefore be combined with the integrals in (8) 

 without altering their values, so that the relations may be 

 Avritten 



(11) ( \j'k{tO)H{ep)de = xf H{td)k{ep)dd = H(tp) - k{tp) 



1 Xfl{qB)H{ep)dd = \fG{qd)k{ep)de^G{qp)-l{qp) 

 which are of the same form as (5); but G^qj)), as will be seen, is 

 the Green's function for the modified problems, and H {tp) bears 

 the same relation to it that H(#j5) bears to G(5'jp). 



3a 



