On the Action of Coupled Circuits. 511 



2. The equations of the problem may be written 



ms S- ns S= * 



Even from the point of view of the student, the method of 

 normal coordinates seems the simplest in this case. For all 

 values of X, 



(LR+X.MS)^-(MR+X.NS)^ = -#+Xy. 



Let 



MR + X . NS=X(LR + X . MS), 

 and then 



(LR + X . MS) ^ 2 (x-\y) + (*-Xy)=0, 



so that the solution is 



*— Xy=*cosf-+eV p 2 =LR + X.MS. 



But the equation for X, 



X 2 .MS-X(NS-LR)-MR=0 . . . (1) 



is a quadratic with two roots Xj, X 2 . The complete solution 

 is therefore given by 



x— \iy = h cos / \-€ x \ x—\ 2 y = h co3 I he 2 j, (2) 



j9i 2 = LR + X! . MS, ^ 2 2 = LR + ^ 2 • MS, 



where 2njp^ 2irp 2 are the periods. Let ZirY^ 27rP 2 be the 

 natural (uncoupled) periods, so that Pi 2 = LR, P 2 2 = NS. 

 Then 



i?1 2 +p2 2 == 2LR + (X 1 + X 2 )MS = LR + NS, 



Pl 2 p 2 2 =LR{LR + (Xj + X 2 )MS} + XjX 2 . M 2 S 2 

 = LR.NS-M 2 RS, 

 or, the coupling being given by M 2 = 7 2 . LN, 



iV+K=Pi 2 +P 3 2 , yiV=Pi 2 P/(i-y 2 )- • (3) 



The method seems so obvious that it is probably well 

 known even in elementary teaching. In a case like this it 

 can be understood when the simple general theory of linear 

 equations with constant coefficients is not familiar. 



202 



