298 PROCEEDINGS OF THE AMERICAN ACADEMY. 



in which x and y are the values of the current in the Circuits III and 

 IV respectively. Eliminating x between these equations we have 



3^ 



(4) {UL, - 3P) + (B.L, + FcM ^ + (^ + ^ + ^3^4^ 



If for brevity we call the left-hand member of equation (4) /(y), the 

 complete solution of (4) is any particular solution of (4) plus the general 

 solution of 



(5) Ai/) = 0. 



Now the general solution of (5) involves exponentials with negative 

 exponents as multipliers, and becomes zero after a few oscillations, so 

 that all we need for the current i/ in case a large number of oscillations 

 are performed, as with a persistent source of waves, is the " steady- 

 state " solution for equation (4). 



In order to get the steady-state solution of (4) let us write, in the 

 place of (4), the equation 



(6) (L,L, - iP) + (R,L, + B,L,) || + (^^^ + ^ + fti?,^0 



which is (4) with sin wt replaced by an appropriate exponential with 

 imaginary exponent. 



Our required solution for (4) can then be obtained from a solution 

 of (G) by getting y from (G), rationalizing the result, taking the imagi- 

 nary part and dividing by i. 



Now a particular solution of (6) is seen to be of the form of 



(7) y— Ye^'"\ 



in which Y is to be determined by substitution of (7) in equation (6). 

 Making this substitution we obtain 



(8) A Fw^ - Bi Yio^ -CYw^ + Di Fa, -f FY = - EMu,\ 

 where A, B, C, D, and F are the coefficients of equation (6). Whence 



- EM J" 



(9) F 



{AiJ - Cuj-' + F) - {Bio' - Dw) i' 



