PIERCE. — THEORY OF COUPLED CIRCUITS. 301 



For any given value of U and the necessary other constants of the 

 circuits, equation (19) enables us to compute the resonant adjustment 

 of Circuit IV, and ecjuation (20) gives the amplitude of the received 

 current at the resonant adjustment. We have thus found the best V 

 and the best Y for a given U. It is proposed next to find what would 

 be the best value to give to U, while also keeping V at its best value, 

 and thus to determine the best possible Y, which we shall call 



-^ max max- 



To obtain the best U we must apply to equation (20) the condition 



dY 



" ->- max ^ 



dU ~^' 



This gives 



or 



(20 a) - R,I\P<J' U-{- B, IP + B^'H, IT = 0. 



Whence, omitting for the present the case of U ^ 0, which is treated 

 on page 302, we have, after dividing by U and transposing, 



(21) Uo,t = ± |/|j(il/V - R,R,), 



in which the subscript " opt " is introduced to designate the optimum 

 value. At the same time V must satisfy equation (19) which com- 

 bined with equation (21) gives 



(22) Vo,t = ± |/|i (J/ V - B.B,). 



lis 



According to the conditions imposed by equation (ld)U'opt and Vopt 

 must either both be positive or both be negative. They cannot have 

 opposite signs. 



Equations (21) and (22) show what values to give Z7and Fin order 

 to obtain the largest possible current (which we shall call Y^naxmax) in 

 Circuit IV. The value of the current, under these conditions, is found 

 by substituting the optimum value of U, namely [/gpt of equation (21), 

 into the equation for Ymax (equation 20). When this is done, we have, 

 after simplification, 



K^'^J -^ max max — / -,._ • 



2 Vi^sRi 



