Pdva/meter Values of Potential Theory. 165 



Pleiuelj's work' is (.(HitiiitMl to the ordinary potential and deals 

 chiefly with the pole \=+l. The present paper extends the in- 

 vestigation to the generalised potential, and also to the general 

 pole Ao. For this charaeteristic number, which may be any what- 

 ever, more general relations are established connecting the resi- 

 dues and the functions H{tj)) and G(pq), which correspond to the 

 modified problems. The boundary discontinuities of these func- 

 tions and their derivatives are investigated, and also certain 

 theorems of reciprocity. Expansions for the various functions are 

 found as power series in the parameter A. 



In the tirst part of the paper the investigation applies to the 

 ordinary and generalised potentials alike. In the second part 

 the ordinary potential is considered separately, and results pecu- 

 liar to Laplace's equation are obtained which depend either upon 

 the fact that A=±l are here characteristic numbers, or upon the 

 special value of the integral of h(tp) extended ovei- the boundary. 

 Values for the boundary integrals of the different functions are 

 investigated. Further from the convergence of the above expan- 

 sions when |Al = l a value is deduced for the conductoi' potential. 

 It will also be shoAvn that the solutions of thti second boundary 

 problem for the inner and outer regions ai'e expressible in terms 

 of a single function. 



Finally the case of the generalised potential is considered 

 separately. The value is found of the integral of Ii(tp) extended 

 over the boundary, in terms of the potential of a space distribu- 

 tion of matter. Further relations are found connecting the boun- 

 dary integrals of the other functions involved. 



I. — Ordindry and (jeneralised poientials. 



^1. Solntions and flieir p(dex. The solutions of the boundary 

 problems as given by (o), when expressed in terms of the resolvent 

 H(^s) become^ — 



(3') \ W(p) ^fi{l)[k{fp) + \rR{te)h{(Jp)d6]df. 



'. V(p)=/[gU>t) + \/\j{p6)K{et)dd]Uf)df 

 where 



h{ep)=fy{ep) 



6 being a point on the boundary, and {/ (p>/) is a particular solu- 

 tion of Laplace's ecjuation if the potential is ordinary, and of the 

 equation (2) if it is genei'alised. The value of this function is 

 given by — 



1 Cf. also " PotentialUieoreti&che Untersuchunfjen," Teubner, Leipzig (1911). 



2 Cf. Weatherbuin. Loc. cit. § 2. 



