Electric Currents in Networks of Conductors. 231 



And if x and y are adjacent meshes, and X H represents the 

 resistance of the common boundary, then — ^R will be the 

 element in the «#th row and yth column, and also in the yth 

 row and #?th column ; but if x and z are nonadjacent meshes, 

 then will be the element in the #th row and zth column, and 

 also in the 0th row and «#th column. Having formed this 

 determinant, which we call the network determinant, we 

 divide it by the first minor of its leading element ; and the 

 quotient is the resistance of the network between the two 

 points, joined by the zero-conductor forming the added mesh. 

 It is seen that, owing to the mode of formation of the network 

 equations, the network determinant is a symmetrical deter- 

 minant — that is, one half of the determinant is the reflection, 

 as it were, of the other half in the diagonal considered as a 

 mirror. 



§ 7. As a means of comparing the results of this method 

 with other known results, let us take the exceedingly simple 

 case of three conductors joining two points in what is com- 

 monly called multiple arc. 



Let 1, 2, and 3 (fig. 6) be the three conductors joining 

 two points A and B ; let their respective resistances be 

 r h r 2j r z \ then join A, B by a dotted line so as to make one 

 added mesh, and let the resistance of this added circuit be 

 zero. Then, without waiting down the equations to the cycles, 

 we see that the network determinant is 



d n — 



n 



— r x 







-n 



n+r 2 



-^2 







—r 2 



r 2 + r B 



The elements r 1} r Y + r 2 , r 2 + r d of the dexter diagonal are the 

 sums of the resistances which bound each mesh, x, y, and e, 

 taking the added mesh x first. 



The other elements of the first row are the resistances, with 

 minus sign prefixed, which separate the mesh x from mesh y 

 and mesh z ; or are common to x and y and x and z, viz. ?\ 

 and zero, because x and z are nonadjacent. And, similarly, 

 if m and n are any two meshes, then the element in the nth 

 row and mth column is the resistance separating or common 

 to the two meshes ; and the element in the nth row and mth 

 column is identical with that in the mth row and nth column : 

 zero being placed as an element if these meshes, m and n y 

 have no common boundary or circuit. 



The above determinant is easily evaluated. By adding the 

 first row to the second for a new second row, and this new 

 second row to the third for a new third row, we transform the 



