242] INDUCTION OF CURRENTS. 501 



we have for the periods, 



/7T _ 27 /77/_ 2<7 



= ^T' * ~~^' 



so that 



r = 

 If we introduce the periods of the two circuits alone, 



T, = 27T N /2iA, T 2 = 27T 



and a quantity 6 which is nearly a mean proportional between 

 them, 



these periods become 



T = 



(9) 



Sp- 



in case Ti = T<i, 



T 2 = T 1 2 + 2 = 47r 2 OK' 1 . 

 (10) /2 _ 2 2 _ 2 V * 



This is a case of so-called resonance, though not the one that we have 

 examined. We see that one of the periods is greater and the 

 other less than the common period of the separate circuits. 



If the period of one of the circuits is much greater than that 

 of the other, so that both 



T! > T 2 and Tf - T<? > 2&>, 



we have, developing the square roots by the binomial theorem, the 

 approximation, 



<9 4 

 a'-av-fais 



(ii) 



In this case the longer period is nearly that of the longer individual 

 period, being somewhat longer, while the shorter period is some- 

 what shorter than the shorter individual period. This is probably 



