Electric Currents in Networks of Conductors. 



235 



say, adjacent angles of the hexagon, in which case no such 

 simple direct method as employed above can be used. 



§ 10. The following example will give a good illustration 

 of Maxwell's method of treating network problems, viz. the 

 case of Sir W. Thomson's resistance-balance for small resist- 

 ances. In this arrangement (fig. 10) 9 conductors join 6 

 points and form 4 cells. B is the battery-circuit in which 

 operates an electromotive force E. Let the four cycle cur- 

 rents be denoted by x+y, y, z, and w. These are the imagi- 

 nary like-directed currents round the circuits, and the real 

 currents in the branches are the differences of these. 



The problem is to determine the current in the galvano- 

 meter branch (x, and the relation of the resistances when this 

 current through Gr is zero. Let P, Q, S, T, R, r, D be respec- 

 tively the resistances of the branches, and Gr the resistance of 

 the galvanometer circuit, and B the resistance of the battery 

 circuit. Then x+y and y being the imaginary like-directed 

 currents in the two adjacent meshes of which the galvanometer 

 branch is the common boundary, then x+y—y=.x is the 

 current through the galvanometer. 



Proceeding to write down the cycle equations, according to 

 Maxwell's rule, we have 



(P + G+Q + R)^+y-G?/-Q£-Rw=0, 



{T + r+S + G)y-Gx~+~y-Sz--rw = 0, 



(Q+S + DU-Sy-Q#+y-Dw=0, 



(R + 0-tr + B)w— Biw+y-Oz— ry=E. 



Rearranging these equations and solving for w, we have the 

 ollowing value : — 



-Q + S, D, -D 



T+S + r, T + r, -r 



P-rQ + R, P + Q, -E 



E 



in which A is the determinant of the four equations in x, y, 

 z, and w, and whose specific value does not concern us. 



This gives the current in the galvanometer-branch ; and if 

 this is zero, then the determinant in the numerator of the equa- 

 tion giving x must be zero. Hence, when x is zero, we have 



-Q + S, D, 



T + S + r, r+% T 



P + Q + R, P + R, P 



= 0, 



