530 PARTICULAR CASES OF INDUCTION. 



The expression ^Mi represents the sum of the products of 

 each coefficient of mutual induction by the sum of the intensities 

 of the current in the two corresponding wires. If there is a 

 magnetic or electrical system external to the first, the corresponding 

 variation of the flow of magnetic force Q in the polygon will also 

 produce an electromotive force. We shall thus have for the 

 contour of the triangle, 



(48) 



Integrating this expression between and /, we get 



' I 



If all the coefficients of induction are constant, and the initial 

 and final intensities are the same in each of the wires, the second 

 term of the second member is null. Finally, if the external work 

 is also null, we get simply 



irdt, 



or if m is the quantity of electricity which passes in a wire of 

 resistance r, and supposing the resistances constant, 



= ^ mr. 



Comparing this equation with that given by KirchhofFs law 

 (211), we see that they are of the same form, and we may 

 deduce from it the following theorems : 



i st. The distribution of the discharges in any given circuit is 

 inversely as the resistances, when the form of the circuit does not 

 change, and the initial and final intensities are the same in each 

 wire, and there is no external work. 



2nd. Even when the former conditions are not fulfilled for all 

 the wires, the equations of distribution do not contain the co- 

 efficients of self-induction of the branches in which the final and 

 initial intensities are the same. 



551. PHENOMENA OF INDUCTION IN TELEGRAPH CABLES. In 

 studying the propagation of electricity in cylindrical conductors 



