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



existence an electromotive force in that branch. Let the 

 internal resistance of the electromotor be included in the 

 quantity B, representing the resistance of the branch A. Then 

 apply Ohm's law to the cycle x formed by the conductors B, 

 I, H ; we have E-Ba?=y-a. 



x is the actual current in this case flowing in the resistance B, 

 and the potential at the ends of B is equal to the effective 

 electromotive force acting in it less the product of the resist- 

 ance of the conductor multiplied by the current flowing in it. 

 For the conductor I we have similarly 



y— £— (*— y)I- 



Hence x— y represents the actual current in I : it is the dif- 

 ference of the imaginary currents flowing round the x and y 

 cycles in the positive direction. And for the conductor H 

 we have also £-a=(*-s)H. 



Add together these three equations, 



E=7— a + B^ ; 



0=/3-y + 0-*,)I, 



and we have, as the result of going round the cycle x formed 

 of conductors B, I, and H, 



B=^(B + I + H)-yI-0H (1) 



a, /3, 7 have disappeared in virtue of these opposite signs. 



This equation (1) is called the equation of the x cycle; and 

 w r e see that it is formed by writing as coefficient of the cyclic 

 symbol x the sum of all the resistances which bound that cycle, 

 and subtracting the cyclic symbol of each neighbouring cycle 

 multiplied respectively by the common bounding resistance 

 as coefficient, and equating this result to the effective electro- 

 motive force acting in the cycle, written as positive or nega- 

 tive according as it acts with or against the imaginary current 

 in the cycle. This is Maxwell's rule. 



Since there are k cycles or meshes we can in this way form 

 k independent equations, and by the solution of these deter- 

 mine the k independent variables, x, y, z, &c. The value of 

 the current in any branch is then obtained by simply taking 

 the difference of these variables belonging to the adjacent 

 meshes, of which the conductor or branch considered is the 

 common boundary. 



§ 3. Let us now consider the most general case possible, in 

 which we have a network composed of linear conductors suf- 

 ficiently far apart to have no sensible mutual induction, and 

 let there be electromotive forces acting in each branch or 



