230 



Dr. J. A. Fleming on the Distribution of 



§ 6. We shall proceed to illustrate this method by a few 

 examples. 



1. Find the resistance between the points 1 and 3 (fig. 5) 

 of a network consisting of five conductors, whose resistances 

 are A, B, C, D, E, joining four points, 1, 2, 3, and 4. 



Connect 1 and 3 by an imaginary conductor of zero resist- 

 ance, and having an electromotive force, e, supposed to act in 

 it. Let at, y, z denote the cycles or imaginary like-directed 

 currents in the three meshes so formed, and write down the 

 current equations, according to Maxwell, for these three 

 cycles : — 



(A + B)tf —Ky — Bz =e, 



-&x + (A + E + D)y -Es =0, 



— Bar 



■By 



+ (B + C-E>=0. 



Then, by what has been shown above, the resistance R be- 

 tween the points 1 and 3 of the network is given by the 



expression 



R = 



(A+B), 

 -A, 

 -B, 



-A, 



(A + E + D), 



-E, 



-B 

 -E 



(B + C + E) 



(A + E + D,) -E 

 -E, (B + C + E) 



In dealing with numerical cases we need no longer intro- 

 duce any notice of imaginary electromotive forces, but proceed 

 according to the following rule. 



To determine the resistance of a network of conductors 

 between any two points on the network. Join these two 

 points by a line whose resistance is supposed zero, and give 

 symbols to the meshes of the network so formed ; calling 

 this additional mesh produced by the added zero conductor 

 the added mesh. Then write down a determinant whose dexter 

 diagonal has for elements the sum of the resistances which 

 bound each mesh, beginning with the added mesh ; and for 

 the other elements of each row the resistances which separate 

 this mesh respectively from adjacent meshes, and having the 

 minus sign prefixed, zeros being placed for elements corre- 

 sponding to nonadjacent meshes. 



More explicitly, if we denote by a 9 y, z, &c. the meshes, 

 x being the added mesh, and by 2R*, 2R^, 2R*, &c. the sum 

 of the resistances which bound each cycle, then these will be 

 the elements along the dexter diagonal of the determinant. 



