328 ELECTROKINETICS. [PT. II. CH. VIII. 



is continuous, in the case of electricity, in passing from the con- 

 ductor 1 to the conductor 2, we have at the surface of separation 



(10) F 2 -F 1= :# 12 , 



where E^ is a quantity depending on the nature of the two 

 conducting substances. 



In the theory of heat, if we have a chain of conductors in 

 contact with each other, surrounded by a non-conductor, we may 

 have equilibrium, but in the case of electricity this is the case 

 only if the sum of the discontinuities of potential is zero, 



(11) E=Eu + E m + ...+E m = 0. 



Conductors may be divided into two classes. Those of such 

 a nature that for any number of them an equation of this sort 

 holds constitute the first class. To it belong all metals (their 

 temperatures being the same). To the second class, for which 

 in general such equations do not hold, belong solutions of salts 

 and dilute acids. 



If we have a set of conductors of either class, the constants 

 E rj r+l being given, and also the conductivity X as a point- 

 function, we shall show that the problem of flow is determined 

 as soon as we are given any two equipotential surfaces. 



Let V A be the potential at one of the surfaces A, V B that 

 at the other B. 



Let <X> be a function holomorphic in the whole space occupied 

 by the conductors, satisfying the differential equation 



and the boundary condition 



at surfaces of separation of two conductors, taking the value 

 unity for all points of the surface A, and the value zero for all 

 points of the surface B, while d<&/dn = for all points of surfaces 

 separating the conductors from the surrounding insulators, or at 

 infinity, if the conductors reach so far. Then if ^ be the potential 

 function in the conductor 1 (in which lies the surface A), v 2 

 that in the conductor 2, . . . v n that in the conductor n (in which 



