554 Mr. Iolo Jones on the Period and Decrement 



The fundamental equations for such a system are 



L i§ +M f +R ^ +v i=° h 



and L/^+M^I+R^O, (2) 



and the current flowing through the condenser is given by 



k=0^. (3) 



Substituting this value of i l in (1) and (2), these equations 

 become 



L 2 ^ + MC 1 < y 2 1 +B 2 ; 2 =0. .... (5) 



Assuming Y t =Ae izt , ■/ 2 = B«*~*, substituting in (4) and (5) 

 and eliminating the ratio A/B, we arrive at the cubic 



(L 1 L 2 -M 2 ): 3 + (L i R 2 + L 2 R 1 )^ 2 + (^ 2 +^1^)* + ??=0, 



M 2 



and on introducing the coupling T T represented by -IP, this 



lJllJo 



cubic becomes 



" + l-tfKU + L,/ + i-FVl,c, + l x • lJ 2 



The solution for V 1 is known to be of the form 



ae'W'cos (2tt at — 6) + he~ ^ , 



where \ represents the decrement and n the frequency 

 .of the oscillations, so that the roots of the cubic (6) are 

 given by 



5f 1= = — k i -\-2irni\ 



~ 2 —~ki—2TTni }> . ..... (7) 



*,= -& J 



For the purpose of tracing the effect on 7^ and y* of varying 

 the ratio R 2 /L 2 a number of numerical cases have been worked 

 out. In these the following numerical values are assumed — 



