150 Prof. E. Taylor Jones *on the Potential 



as one coil. The ratio o£ the squares of the frequencies in 

 these cases is 



n 2 1-P 



(16) 



a result which suggests an experimental method for deter- 

 mining the coupling. 



In general, if we write u for the ratio LjOj/L^C^ the- 

 frequency-ratio is given by 



n 2 2 _ s + u+ \f~(s + u) 2 — ±{l-k 2 )u 



For any given value of k 2 the frequency-ratio is least 

 when u = s, i. e. when L 1 Ci = 5L 2 C 2 . 



In order to find the amplitudes multiply (11) by any 

 factor X and add its terms to those of (9). We then have 

 the equation 



{L ] +\(L 1 + L 2 0}C 1 ^ + {(L 1 + L 12 ) + X,L 2 }C 2 ^ 2 



+ V 1 + XV 2 = 0.. 

 If X is so chosen that 



(Li + L^+XsL^C^MLx + XlLi + LaOfC!, . (1«> 

 then 



{L 1 +X(L 1 +L 21 )}O t ^(V 1 +\V 2 ) + (V 1 +\Y 2 ) = 0. (19) 



The two values of X, viz. Xi and X 2 , may be calculated 

 by (18) in terms of the coefficients of equations (9) and (11). 

 They may also be expressed in terms of the frequencies n t 

 and n 2 , for, by (19), 



i -^- 2 ={L 1 + X 1 (L 1 + L 21 )}C 1 , 



^i- 2 = {Lx + ^Li + L^jd. 

 Thus 



Xl -(L 1 + L 21 )C 1 (^rW~ LlCl ) 



(20) 



The solution of equation (19) is represented by the two 

 normal vibrations 



V 1 + A. 1 V 2 = A 1 sin(27r^ + a i ), 1 



V 1 + X 2 V 2 = A 2 sin (2ir ni t + 8 2 ). J ' " ' ^ J V 



