332 ELECTKOKINETICS. [PT. II. CH. VIII. 



maining parts X3F/3ft = 0. Consequently the integrals over the 

 bounding surfaces disappear, and we have 



9 A, 9F M J 



s ( x TZ )\ dT 

 asF /9SF 



In order for J(V) to be a minimum, this must be positive 

 for all possible choices of the arbitrary function SV. This can 

 be true only if we have everywhere in the region r 



and at every surface of discontinuity S, 



Consequently the statement that J is a minimum is equiva- 

 lent to stating that q is solenoidal. 



166. Integral form of Ohm's Law. We have seen in 35 

 that the solenoidal condition signifies that the flux, 



/= q cos (qn) dS 



across any surface bounded by the sides of a vector tube is the 

 same for all parts of the tube. In the case of electrical flow, 

 the flux is called the current (current-strength, or intensity) in 

 the tube. Although V has discontinuities, the function <I> has 

 not. Since we have between any equipotejitial surfaces A 

 and B, 



the ratio of the flux to E+V A V B , the difference of potential 

 plus the sum of the sudden rises of potential as we go in the 

 direction of flow, thus depends only on the function <3>, which 

 depends only on the configuration of the space r, and the values 

 of the function X. That is, the ratio of the flux in any tube of the 



