Electric Currents in Networks of Conductors. 229 



Now consider that part of the network which remains if the 

 conductor B is removed, and let us imagine that a current n 

 continues to be forced into it at 7 and drained out at a ; the 

 total resistance of that part of the network, not counting B, is 



but this is equal to 



y — a, 

 I-B. 



X 



Now since the resistance of B maybe anything, let it be zero; 

 then the total resistance of the network between 7 and a will be 



E 



R= 



x 



but 



TEA^n 



L A w Jb=o, 



where the suffix and bracket denote that after the determi- 

 nants are formed from the cycle equations, according to 

 Maxwell's rule, then in them B is put equal to zero. 



If we denote the determinant of all the n-cycle equations 

 under the condition of B = by d n , and by d n - Y the first minor 

 of this or the minor of its leading element corresponding to the 

 coefficient of x with the resistance of the circuit containing 

 the effective electromotive force put equal to zero, we have 

 for the total resistance R of the network between the points at 

 which the current enters and leaves, the expression 



d n 



R= 



d n - 



Since, then, as we have seen, the linear equations for the 

 cycles can always be solved by evaluating the determinants, 

 it follows that in all cases, no matter how complicated, the 

 resistance of any network can be calculated by simple arith- 

 metic processes from the given resistances of the branches or 

 conductors which compose it. We have therefore an interest- 

 ing extension of Maxwell's method of calculating the currents 

 in a network and the potentials at the junctions to a method 

 of calculating the combined resistance of a number of con- 

 ductors forming a network ; which method consists, as seen 

 above, in forming a certain determinant whose elements 

 are formed of the separate resistances of the branches, and 

 dividing this determinant by another of an order next below, 

 viz. the first minor of its leading elements ; and we find 

 that the resistance between any two points of any network 

 of conductors, however complicated, is expressible as the 

 quotient of a certain determinant by another formed from it. 



