Electrostatics and the Steady Flow of Electricity. 579 



(B) At the surface of separation, S, of conductors of 

 different conductivity X the normal derivative is discontinuous 

 according to the formula 



*-£(*-)-*£ (.+)-0. 



The conductivity X is positive and will be assumed constant 

 for each conductor. 



(0) At the surface of separation, ®, between a conductor 

 and a non-conductor the potential is continuous, while its 

 normal derivative vanishes on the side of the conductor. 



The first of these conditions, (A), is clearly satisfied by 



the potential u { (p) of a double stratum of moment — t] (s) 

 over S, expressed by w7r 



id (p) = \r)(s)h(sp) ds. 



In order to satisfy both (A) and (0) we observe that Ui{p) 

 is continuous at the surface © separating conductor and 

 non-conductor, so that it has a definite normal derivative 

 there. We may then determine a solution u 2 (p) of (2) 

 having at © on the side of the conductor a normal derivative 

 equal and opposite to this. Such a function n 2 {p) is, in 

 virtue of (3) and (4), expressed by 



? <2(» = iG+: 



ai* 1 



The sum u (p) = u i(p) + l h{p) 



then satisfies both (A) and (C). 



In order to make provision for (B) express the final 

 solution in the form 



V(j>)=u(p) + VS(p). 



Then TJ{p) must be a solution of (2) which is continuous at 

 the surface S, while it satisfies (G), for both V and u do so. 

 Substituting this value of V in the boundary condition (B) 

 we find, since u(p) is continuous at 2, that 



^^-^(^(XW)^), (23) 



u(o) being a known function of the point a. 



As a discontinuous normal derivative is characteristic of a 

 simple stratum let us endeavour to satisfy (23) by a simple 



stratum potential of density } - ix(a) over 2, that is 



r(p) = §9(pa)n{?)d<r. 



2 P 2 



