322 Mr. W. P. Boynton on the 



Substituting the values of A 2 from equation (20), and of 

 a and 7 from equations (10) and (11), or directly from 

 equations (2) to (7), rationalizing, and performing the 

 necessary algebraic simplifications, we get 



f" Yq 2 M 2 K 1 2 (R 1 L 2 + R 2 Li) 



J o v 2^- 2 [R 1 R 2 (L 1 K 1 -L s K 2 ) 2 + M 2 (R 1 K l + R 2 K 2 ) 2 J' 



The " effective " potential squared will be this quantity 

 multiplied by 2n, where n is the frequency of the alternating 

 current charging the condenser ; or, calling this V 2 2 , 

 y - 2 _ nY 2 M 2 K 1 2 (R 1 L 2 + R 2 L 1 ) 



* 2 ""R 1 B 1 (L l K 1 -L s K,) 8 +M»(R 1 K 1 + R,Ki)»' * ^ 0) 

 The general expression for the current in either circuit is 



T — d Q- e ~ at { (-«A + /8B) cos j3t + (-«B-/3A) s'm/3t\ 

 dt~ +c-^(- 7 C + 5D)cos^ + (-7D-SC)sin&} *' 



\2 dt = (-<*A + /3B) 2 +(aB + /3A) 2 + (- 7 C + gP) 2 +( y D-j-gC) 2 

 4a 4 7 



_ (a 2 + /3 2 )(A 2 + B 2 ) ( 7 2 -i-S2)(C 2 + D'Q 

 4a + 4y ' 



which becomes, neglecting a 2 , y 2 , B 2 , and D 2 as small, 

 fl Vl «£*!+«? 



Jo 4a - *7 



Applying this to circuit 2, where A 2 2 =C 2 2 , and substituting 

 and reducing as before, we get 



K 1 KoV 2 M 2 (R 1 K 1 -fR 2 K 2 ) 



.1 



J. 



\ 2 *dt = 



2[R 1 R 2 (L 1 K 1 -L 2 K 2 ) 2 +M 2 (R 1 K 1 + R 2 K 2 ) 2 ] ? 



a?^d the " effective " current squared is 



T -s _ nV 2 M 2 K 1 K 2 (R 1 K 1 + R 2 K 2 ) 



h _ R 1 R 2 (L 1 K 1 -L 2 K 2 ) 2 + M 2 (R 1 K 1 + R 2 K 2 ) 2 ' ^ 



In the case of the primary circuit we shall see that with 

 our arrangement the coefficient C 1 decidedly preponderates 

 over the others. Then we have 



1 



4 r 



(VIXRiK! + R 2 K 2 ) ((LjKi - L 2 K,) 2 + 4M 2 K 1 K 2 ) 



-(L i K 1 -L 2 K 2 )(R,K 1 -R 2 K 2 ) V(L 1 K 1 -L 2 K 2 ) 2 + 4M 2 K 1 K i! ] . , 2 j 

 4K 1 K s LRiR 2 (L 1 K,-L 2 K s ) 8 + M 2 (R,K I -R 3 K 3 ) 4 ] 



and I, 2 is this expression multiplied as usual by in. 



