314 Steady Currents in Linear Conductors [OH. ix 



by a conductor, we must simply suppose R PQ to be infinite. Discontinuities 

 of potential must not be excluded, so we shall suppose that in passing through 

 the conductor PQ, we pass over discontinuities of algebraic sum E PQ . This 

 is the same as supposing that there are batteries in the arm PQ of total 

 electromotive force E PQ . We shall suppose that the current flowing in PQ 

 from P to Q is X PQ , and shall denote the potentials at the points 1, 2, ... by 



K,K, .... 



The total fall of potential from P to Q is V P V Q , but of this an amount 

 EPQ is contributed by discontinuities, so that the aggregate fall from P to 

 Q which arises from the steady potential gradient in conductors will be 



Hence, by Ohm's Law, 



V P - VQ + E PQ = R PQ X PQ . 



If we introduce a symbol K PQ to denote the conductivity -p , we have 



-tipQ 



the current given by 



(289). 



Suppose that currents X l ,X 2 ,... enter the system from outside at the 

 points 1, 2, ..., then we must have 



since there is to be no accumulation of electricity at the point 1, and so on 

 for the points 2, 3, .... Substituting from equations (289) into the right 

 hand of this equation, 



.) + K m E + K l3 E ls + ............... (290). 



The symbol K PP has so far had no meaning assigned to it. Let us use it 

 to denote (K Pl 4- K P2 + K^ +...); then equation (290) may be written 

 in the more concise form 



X 1 = -(KV 1 + K,&+...) + K 1 A + K K E li + ......... (291). 



There are n equations of this type, but it is easily seen that they are not 

 all independent. For if we add corresponding members we obtain 



X, + X, + . . . + X n = -XK (- + KV + . . . + K ln ) + 22 (K PQ E PQ + K qP E QP ). 



The first term on the right vanishes on account of the meaning which has been 

 assigned to K u , etc.; while the second term vanishes because E PQ = E QP , 

 while K P Q = KQP- Thus the equation reduces to 



X 1 + X 2 +...+Z n = 0, 



which simply expresses that the total flow into the network is equal to the 

 total flow out of it, a condition which must be satisfied by X l} X 2> ... X n at 



