PIERCE. — THEORY OF COUPLED CIRCUITS. 299 



or writing the real part of the denominator as P, the imaginary as Qi, 



F-qi 



Rationalizing equation (11), taking only the imaginary part and 

 dividing by /, we have 



_ — EMoi^ (F sin wt + Q cos wt) , 



y- F^+q" 



— EMiJ^ sin ( o,t + tan-l-^^ 

 (12) = ^ ^^ 



VF' + q^ 



Replacing F and Q in the denominator by their values in terms of the 

 constants of the circuits, we have 



Equation (13) is a well-known solution of equation (4), and gives 

 the value of the current y in Circuit IV after the effect of the free- 

 period initial disturbance has subsided. It is seen that the current y 

 in Circuit IV is sinusoidal, with the frequency of the e.m.f impressed 

 on Circuit III. We shall concern ourselves only with the absolute 

 value of the amplitude Y of this current. 



Dividing numerator and denominator of equation (13) by u? and 

 factoring, we obtain 



(14) Y^ 



EMw 



Now let 



(15) Z7=Z3W-7^, F=i/4a)-— -• 



