Electric Currents in Networks of Conductors. 233 



The dexter diagonal has for each element 4, viz. the sum of 

 the four resistances, each to unity, which form each mesh or 

 cell. And all the other figures, say, in the nth row, are the 

 resistances (with minus sign prefixed) separating the nth 

 mesh from all other meshes, zero being placed in the column 

 corresponding to any mesh which has no common conductor 

 or branch with this nth mesh. The order in which the columns 

 stand and also the rows correspond to the order in which the 

 meshes are numbered in fig. 7. 



The numerical value of this determinant is easily found to 

 be 288 = 3 x 96 = d w . Now if we take the first minor of its 

 leading element, we get a determinant formed of the elements 

 included in the dotted rectangle ; and taking this as a separate 

 determinant and evaluating it, we have its value 



<4-i = 192 = 2x96; 



hence the resistance of the network between the points A 

 and B is 



d n 288 ., , .. 

 ^ = l92 =1 * UUlts - 



§ 9. One more simple numerical case may be taken and 

 compared with the results of know T n methods. 



Let a hexagon of conductors be taken (fig. 8) having 

 crossed diagonals all meeting in the centre. Let the resistance 

 of each side, as ab, be unity, and also let the resistance of each 

 semidiagonal, as Oa, be unity. ■ Then required the combined 

 resistance of this network of 12 conductors between the points 

 A and B diametrically opposite. Join the points A and B by 

 a dotted line of zero resistance, making an added mesh 1. 

 Mark the other meshes 2, 3, 4, 5, 6, 7. Then by forming the 

 network equations it is easily seen that the network deter- 

 minant d n is 



3 -1 -1 -1 



-1 3 _i o-l 



-1-13-1000 



-1 0-1300-1 



0-0 3-1 



0-1 3-1 



0-1 0-1 3 



= d n . 



The value of this determinant is 256. 



