570 Mr. C. E. Weatherburn : Problems in 



on the boundary, and t~ the corresponding point of the outer 

 region. It was shown that the required solutions may always 

 be obtained by means of the exponential potentials of simple 

 and double strata over the boundary, the results being 

 expressible in the form * 



W(i»;=j , /WH(<p)#i« 



Y( P ) =]Qipt)f(t)dt fi »•■••<>; 



where H and G are connected by the relations 



mtp) - h(tp)=\ jJK(t$)h(Sp)d$ = \$7i(tS') 1 B.(Sp)d$, I a. 



G(^)-?(<7p)=\j v ^ 



9 (QP) being the function - — • of the distance r between the 



points p and q, h(tp) the normal derivative of g(lp) at the 

 boundary point t, and H(ts) the solving function or resolvent 

 of h(ts) regarded as the kernel of an integral equation of 

 Fredholm's type. 



Part I. — Electkostatics. 



§ 1. First Problem. — In a system of conductors acted 

 on by given fixed external charges the potentials of the 

 individual conductors are known. Required to find the 

 distribution of electricity. 



The number n of such conductors is immaterial. Their 

 surfaces form the multi-connex boundary between the internal 

 and external regions. The potential P* (t=l, 2, ... , n), is 

 constant throughout each conductor, being zero if the con- 

 ductor is earth connected. Assume the most general dis- 

 tribution of electricity, viz. volume and surface distributions 



of densities — 7j(p) and ^- /i(t) respectively. Then by the 



extension of Poisson's theorem it follows that 



A*F i -k*F i =-2 Vi (p), 

 and hence 



v^jfi, n 



so that the volume distribution has a constant density for 

 each conductor. 



* Loc. cit. First paper, §2, (12). Cf. also another paper by the 

 author, " Singular parameter values in the boundary problems of the po- 

 tential theory," Proceedings of the Royal Society of Victoria, vol. xxvii. 

 Part 2 (1915), pp. 164-178. 



