Electricity in a Network of Conductors. 281 



tremities remain unchanged. For currents will enter any one 

 mesh of the network at certain points and leave it at certain 

 other points. One of the former points must be the point of 

 maximum potential in the mesh, one of the latter the point of 

 minimum potential. The circuit of the mesh, therefore, con- 

 sists of two parts joining these two points, and to any point 

 in one of the parts will correspond a point of the same poten- 

 tial in the other part. We may therefore suppose every point 

 in one in coincidence with points of the same potential in the 

 other ; that is, the mesh replaced by a single wire joining the 

 two points, and such that the currents entering or leaving it 

 by wires joining it to the rest of the system and the potentials 

 at the points of junction, are not altered. 



Since the only electromotive force is in the wire AB,, the 

 current must enter the network at one of its extremities, A, 

 say, and leave at the other extremity ; A and B are therefore 

 the points of maximum and minimum potential of the network. 

 Hence we can replace the meshes of the system one by one by 

 single wires, keeping CD unaltered until we have reduced the 

 network to two meshes, one on each side of CD, connected, if 

 necessary, by single wires to A and B respectively. Each mesh 

 and connecting-wire can be replaced by two wires joining A 

 and B respectively with C and D, and thus the whole system is 

 reduced to an equivalent system of the form shown in the figure. 

 We can now deduce from this simple system relations for the 

 currents and potentials in the conductors AB, CD, which will 

 hold for these conductors in the more complex system. 



Let the electromotive force hitherto supposed acting in AB 

 be transferred to CD, while the resistances r 5 , r 6 are main- 

 tained unaltered. The value of y 6 will be obtained from (6) 

 by retaining the numerator unaltered and interchanging 

 r 5 and r 6 , r x + r 2 and r Y + r s , r 3 + r 4 and r 2 + i\ in D. But 

 these interchanges will not effect any alteration in the value 

 of D ; and hence the new value of 7 6 is equal to the former 

 value of 75. Hence the theorem : — An electromotive force 

 which, placed in any conductor Of of a linear system, causes 

 a current to flow in any oilier G p would, if placed in C p , 

 cause an equal current to flow in G[. 



If the arrangement is such that when the electromotive 

 force is in C/ the current in C m is zero, the current in 

 Cj will be zero when the electromotive force is in G m ; and no 

 electromotive force in one will produce a current in the other. 

 The two conductors are in this case said to be conjugate. 



We can easily obtain another important theorem. The 

 five conductors AC, AD, BC_, BD, CD in the figure may be 

 regarded as the reduced equivalent of a network of con- 

 ductors, at one point of which, A, a current of amount y 6 



