Electrostatics and the Steady Flow of Electricity. 569 

 as a limiting case, as is clear from the relation 



In the present paper will be discussed various problems 

 of electrostatic equilibrium, and the general problem o£ the 

 steady flow of electricity under the exponential potential 

 e~ kr /r. The data in the statical problems may assume 

 different forms. The potentials of a system of conductors 

 may be given, or only their total charges; while the potential 

 in the external field may be continuous along with its deri- 

 vative, or may experience at certain surfaces a discontinuity 

 of normal derivative such as that expressed by equation (15) 

 below. The distribution of electricity on the conductor will 

 not in general be confined to the surface; but will consist 

 of a volume distribution of density rj and a surface distribu- 

 tion of density fi. The former is related to the potential by 

 an extension of Poisson's theorem proved elsewhere*, viz. 



A2V-PV=-47T7;, (1) 



from which it follows that, since the potential is constant 

 throughout a conductor, 77 is also a constant, different for 

 each conductor. The surface distribution /x over the con- 

 ductor possesses exactly similar boundary properties to those 

 of simple strata under the ordinary potential. 



In earlier papers f the author has considered various 

 boundary problems for the equation 



A 2 V-PV = 0, (2) 



which is the equation satisfied by the exponential potential ; 

 in particular those were discussed which correspond to the 

 generalized problems of Dirichlet and Neumann, requiring 

 the determination of solutions W(p) and V(p) of (2) 

 satisfying respectively at the boundary the relations 



X 



f(t))b. 



(3) 



§[W(« + )-W(r)] _ ] [W(*+)+W(r)] =/(«) 



IrrfV. . dV ...1 XrdV. . dV...-[ 



where \ is an arbitrary parameter, t + a point of the inner 

 region indefinitely near the point t of the boundary, but not 



* Weatherburn, " Green's functions for the equation a 2 m-Fm = 0,&c." 

 § 5. Quarterly Journal of Math. vol. xlvi. (1915). Cf. also Neumann, 

 loc. cit. S. 70. 



t Quarterly Journal of Math., vol. xlvi. Two papers. 



