330 ELECTROKINETICS. [PT. II. CH. VIII. 



By Green's theorem this is equal to 





-m- 



The surface integral is taken over the surfaces A and B, and 

 the surfaces bounding the composite conductor, the integrals over 

 the surfaces separating two conductors vanishing in virtue of (13). 



But at the surface A, <J> X and <f> 2 are both equal to 1, hence 



3>i-<*> 2 = 0, 



and at the surface B, <X>! and O 2 are equal to 0, while on the remaining 

 surfaces d^/dn 9<E> 2 /9?i = 0. Consequently the surface integrals 

 vanish. But the integrand in the volume integral vanishes in 

 virtue of the differential equation satisfied by both functions. 

 Consequently the integral J vanishes, but as in Dirichlet's demon- 

 stration this can only be if <E>j <I> 2 is constant. But since 4^ 

 and <E> 2 are equal on the surfaces A and B, they must be every- 

 where equal. Consequently the solution is unique. 



165. Properties of Vectors obeying Fourier-Ohm Law. 



The vectors F, the electrostatic force, and q the electric current- 

 density are typical of a class of pairs of vector-functions of frequent 

 occurrence in all parts of mathematical physics, distinguished by 

 the following properties. The first vector is lamellar, the second 

 is solenoidal. In isotropic bodies the vectors have the same 

 direction, and their ratio depends only on the physical nature 

 of the body at each point. When two vector-functions have these 

 properties we shall say that they satisfy the law of Fourier- Ohm. 

 The study of the properties of such vectors is of great importance. 

 We shall in general call the solenoidal vector the yte-density, 

 and the surface integral of its normal component over any surface 

 the flux through that surface. 



It is remarkable that the characteristic properties of such 

 vector-functions are embodied in the single statement that if V, 

 the potential function of the lamellar vector, is uniform, finite, and 

 continuous, in a certain region r, its first derivatives possessing 



