172 C. E. Weatherbuim : 



ii. If p is a point of the iuiier region, </ of the outer, and t a 

 point on the boundary. 



(24) (1+A)G(;.^)=(1-X)G(^/.) 



G(tp) = {l-\-X)G{pt) 

 G{t^)={l-k)Q{^t) 

 From (23) and (6) we deduce immediately that if p and q are 

 both in the same region, or both on the boundary, 



(25) ( Q{p^) = QU/p) 



If, however, p and ^ are in the inner and outer regions respectively, 

 we find on substituting from (G) in the first of (24), multiplying by 

 Aq— A and putting A=:Ao 



(26) ^{\+X,)Q(p^) = {l-X,)q{^p) 



I (1+a)6'(m)=(1 -^)G(^p)+j^^-Q(n') 



Similarly from the second and third of (24) we find 



(27) ( Q{tp) = (l+X,)Q(pf,) 

 I Q(^.;)==(l-A„)Q(./0 



and thence 



(28) S 0{fp) = {\ +X)G(pt)-X,q(pt) 

 ( G{t^) = {l-k)G{^/t)+X,Q{^/t) 



II. — The ordinary pofeiti led. 



§6. — Integral Relatione. — The preceding properties are common 

 to ordinary and generalised potentials. We know, however, that 

 while the values A:= + l, Avhich correspond to the problems for the 

 inner and outer regions separately, may both be characteristic 

 numbers for the ordinary potential, they are nofl singiilar for the 

 generalised. The properties arising from the existence of these 

 poles are then peculiar to the ordinary potential. Other special 

 relations arise from the fact that foi' this potential the function 

 h(tp) satisfies ihe integral relation2 



(29) /h{tp)dt=2, 1, or 



according as p is within the closed surface, on the l)oundary or 

 outside, and the integration is extended over the boundary. W© 

 shall find furthci- on a cori-esponding formula for the generalised 

 potential from which tliis may be deduced by putting A;=0. 



Let us suppose that tiie boundary consists of m independent 

 surfaces each possessing at every ])oint a definite tangent plane 

 and two definite principal radii of curvature. The value A=l 



1. Weatherburu. Loc. cit., § 3. 



2. I'lemelj. Noc. cit., S. ^41-4. Another proof is 1).\ (Jrecn's TheoriMii as in S i) of this pujier. 



