Oscillations in Three Coupled Electric Circuits. 39 



From equations (7) i., iii., v., vii. we obtain the following- 

 equation for ft> 2 : — 



g> 2 6 (2M 12 M 23 M 31 - L^L, + Lfc 2 + L 2 M 31 2 + L 3 M 12 2 ) 

 ,/ Li L 2 L» BA RAR 3 RA 



_ft>2 Vc 2 C3 + o 1 c 3 + c 1 o 2 + ~c7- | '^Jr + _ c7/ 



+ PPTT = 0. . . . (8) 



If now we make use of tlie abbreviations given below > 

 equation (8) can be expressed in a slightly more compact 

 form, containing only the couplings and the constants of the 

 separate circuits as coefficients : 



~ LA' 



m2 _ 1 



L 2 CY 



1 



11 "L 3 C 3 





z. _ % 



k2 ~ u 



Z ^ 3 

 ^3 — T~* 



-L<3 



Remembering the values of a, /3, and 7 given in equa- 

 tion (3), we can write equation (8) in the form : 



a> 2 H (l-a 2 -/3 2 - 7 2 -2a/3y) 

 - co 2 4 { (1 - a 2 ) ,i 2 + (1 - /3 2 ) P + (1 - ? 2 )m 2 + &A + & 2 £ 3 + Mi} 

 4- w 2 2 (Z 2 m 2 4 m 2 n 2 + ?i 2 / 2 + M 2 n 2 4- W 2 + & 3 &i™ 2 ) 

 -Z 2 w 2 n 2 = 0. (9) 



This is a cubic equation in co 2 2 , of which the three (real) 

 roots may be termed a^ 2 , oj 2 2 , and o) 3 2 . The formal solution 

 of equation (9) presents considerable difficulties. In any 

 actual case in which the numerical values of the various 

 coefficients are known, the roots may be obtained simply by 

 graphical methods. As will be shown later, equation (9) 

 becomes more tractable if certain simplifications are applied. 

 Solutions are thus obtained for certain cases rather less 

 general than that represented by equation (9). 



