Coefficients of Mutual Induction. 123 



denoted by 7, the coefficient of self-induction of the coil s by 

 L, and the potentials at the points A and E by A and E 

 respectively. 



When the battery-current has attained its steady value, it 

 is evident that the currents x and y will both be nothing, and 

 therefore that A = E, and that the charge of the condenser 

 will be Gyr. But if there has been no throw of the galvano- 

 meter-needle, the average value of the current y during the 

 whole time of establishment of the battery-current has been 

 = 0. Consequently, the total quantity conveyed by the cur- 

 rent x has been equal to the charge of the condenser, or 



1 



xdt = Gyr. 



But, if the average strength of the current y = 0, the average 

 difference of potentials A— E = 0, and the effective electro- 

 motive force in the conductor p is that due to the mutual 

 induction of the coils P and s (for the integral value of the 

 electromotive force of self-induction must vanish). Hence 



I xdt = — 1 — dt = Gyr, 

 Jo P Jo dt 



M = Cpr. 



In order that the galvanometer-current y may be zero at 

 every instant, as well as on the average, during the establish- 

 ment of the primary current, it is essential that the coefficient 

 of self-induction, L, of the coil s should be equal to the 

 coefficient of mutual induction M. This may be proved 

 as follows: — Since, in the case supposed, we have always 

 A — E = 0, we may write 



But x=y—z (since y—0, always), consequently the instan- 

 taneous value of the current x is 



-i't(M-L)g + tgj, _ 



and the simultaneous charge of the condenser is 



Grz= r^ = i[(M-L)7 + L/], 



or 



(M — L)7= {Gpr — L)z. 



But, since it has been already proved that ~M. = Gpr when the 



K2 



