Electric Currents in Networks of Conductors. 225 



conductor. Let the system be considered to have arrived at 

 the steady condition. Let x y y, z, &c. be the cyclic symbols 

 or measure of the imaginary current circulating counter- 

 clockwise round each mesh. Let A, B, C, &c. (fig. 3) be the 

 resistances, and e^ e 2 , e B , &c. the electromotive forces acting 

 in each branch. These are reckoned positive when they tend 

 to force a current round the mesh counterclockwise, and 

 negative when they act in the opposite direction. Then the 

 equation to the x cycle will be 



x(A + J + L) — yj + Oz + Ou -f Ow= e x . 

 The symbols of all the cycles are written down, putting in 

 those of Zj u } and iv with zero coefficients, as they are not 

 adjacent cycles to that of x. We shall have five equations 

 similar to the above for the other cycles, y, z, w, and u. 



Now it can very simply be shown from the theory of deter- 

 minants, that if there are n linear equations of the type 



a 1 x 1 + a 2 x 2 -\ r a n x n =Pv 



b 1 x 1 + b 2 x 2 + bnx n =p 2 , 



k 1 X 1 + k 2 X 2 + K^n^Pn) 



the solution for any variable x 1 is the quotient of the deter- 

 minants 



Pl 



a 2 



P2 



h 



P« 



k 2 



#! = 



(l\ Ct 2 



i£\ fc 2 





The only difference between the numerator and denominator 

 is that the solution for x n is given by writing as numerator the 

 determinant of the n equations having the column^, p 2 . . .p n 

 substituted for its nth column, and then writing down as 

 denominator the determinant of the n equations simply, 



