228 Dr. J. A. Fleming on the Distribution of 



And a similar series of operations reduces this to 



76 5 



21 -2 

 which is equal to 



-76x2-5x21=-257. 



Accordingly a series of simple subtractions and multiplica- 

 tions will effect the evaluation of any numerical determinant, 

 and enable us to solve a series of linear network equations for 

 the currents in all the branches when the numerical values of 

 the resistances of the conductors are given. The equations as 

 written above give as solutions the values of the cyclic sym- 

 bols or imaginary currents round each mesh. To obtain the 

 actual current in any branch, we should have to obtain the 

 values of the cyclic symbols or imaginary currents, for the 

 adjacent meshes of which the given branch is a common 

 boundary. Maxwell ingeniously saves labour in this opera- 

 tion by taking as the symbol for one mesh say a+y, and for 

 an adjacent mesh y (fig. 4), and then the real current in the 

 branch AB is #+y— #=#. 



And the simple rearrangement and solution of the network 

 equation gives at once as value for x the current in the resist- 

 ance AB, which is the common partition of the two meshes. 



§ 5. Returning now to the case when there is only one 

 impressed electromotive force in one branch, we see that in 

 forming the cycle equations only one will be equated to an 

 electromotive force, viz. the equation for the mesh containing 

 the impressed electromotive force in one of its branches. All 

 the other equations will be equated to zero ; and accordingly 

 the equation for the current in any conductor will be of the 



form _ EA fl ,! 



where A n is a determinant of the nth order, and A. w _! is a first 

 minor of this. Referring to fig. 1, we see that, by writing 

 down the five equations of the cycles os } y, z, u, w, we obtain 

 equations by which to calculate the currents in any of the 

 thirteen branches, and the current in branch B will be 



~~± ; 



where A n is the determinant formed of the coefficients of the 

 five equations, and A w _! is the first minor corresponding to 

 the coefficient of x in the equation of the #-cycle. 



We also saw that if y and u are the potentials at the ends 

 of the branch B, j] _ g^ _. _ u 



