LAWS OF BRANCH CURRENTS. 527 



L, and the more rapid is the variation in strength ; when the 

 strength is decreasing the reverse is the case. The effect is the 

 same in the general case, in which none of the coefficients is zero, 

 and equation (46) shows that each conductor behaves for an 

 increasing current, as if it had a greater resistance than for the 

 permanent state. 



Let m and m' be the quantities of electricity which traverse 

 the two conductors in the time /, we have 



rm 



- r'm' + [U - LV + M (i ' - 1)] J = . 



In the case of a discharge, the initial and the final strengths 

 are zero ; if the coefficients L, L', and M were constant, the term 

 within the parenthesis is null, and we have 



rm r'm' = Q. 



It follows that the total quantities of electricity were divided 

 between the two branches according to the ordinary law, although 

 at each instant the division was different; there is therefore ulti- 

 mately compensation. 



This conclusion is only valid when the two circuits have 

 produced no external work ; and particularly that during the 

 discharge no magnet shall have been moved near one of them. 

 Hence, in measuring discharges, it is difficult to use a galvanometer 

 with a shunt. 



548. If the principal current is sinusoidal with the amplitude 

 I , the two branch currents will ultimately have the same periodic 

 character with an unequal difference of phase. By calculations 

 which are analogous to the foregoing, we shall find the following 

 relations between the amplitudes A and A', and the difference of 

 phases < and </>', 



A 2 A' 2 i; 



Lr'-LV+M(r-r') 



27T 

 tan 27rc = 



+ (L + L'-2M)(L-M) 



