300 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Then equation (14) may be written 

 (IG) Y= ^^^- 



This is another form of the expression for the current in Circuit IV, 

 after the effect of the free-period initial disturbance has subsided. 

 From equation (16) it can be seen that Ls and Cz enter in the equa- 

 tion only in the form U ; Lt and C4 enter only in the form V. 



In seeking the dependence of Y on the constants of the circuits let 

 us now follow mathematically the steps taken in the experiment above 

 described. 



First, we shall take a fixed value of U and determine what value of 

 V makes Y a maximum. This value of V that makes Y a maximum 

 we shall call the "resonant value of V." The resonant value of F is 

 obtained by making 



(17) 1^=0. 

 This gives 



(18) {[/V- MW - B,R,) U + (7?4 U + ^3 F) ^3 = 

 or 



(19) V= ^^^-^^ - 



Equation (19) gives the value of V for resonance with any partic- 

 ular given value of U. This relation will be further examined in a 

 later section. 



By the use of equation (19), when we have a given fixed value of U 

 we can calculate the value of V for resonance. The current in Circuit 

 IV for this resonant value of Y may be obtained by substituting the 

 value of F given by equation (19) into the equation for F (equation 

 (16)). If we designate this resonant value of Fby Ymax-, we have 



^{ jj^t% ~ ^'"' " ^^^0' + (^*^+ 



RMWP 



u 4- i?3^ 



EMii. 



Factoring the denominator of this expression, we have 

 (20) ^^^"^ 



V — / R T\r 



-' max — / -"3-^'-' 



^2+y>^2+i?4lV/4^+ U'^ 



^^vrs 



