Oscillations in Three Coupled Electric Circuits. 45 



The solutions for the cases in which l = m, j3 = y, and 

 1=71, ol = /3 can be obtained from (18) by a cyclical inter- 

 change of the letters. 



Eliminating the contribution of the third circuit from (18) 

 we find the roots become 



l2 + m 2 + V0 2 -m 2 ) 2 -4a 2 Z 2 m 2 

 Wi =- 2(1-.-) 



9 _ Z 2 + m a - y/(^-m 2 ) 2 -4a 2 / 2 m 2 

 6,2 -~ 2(l-a s ) 



the well-known result for the case of two undamped coupled 

 circuits. 



Using equation (17) in the abbreviated form 



_ «o /-( f+ff \ 



-A ( f \ 



" 2 - C0 2 \f+g + h)' 



we can now write the form which the general solution 

 assumes for the special case here considered and for the 

 initial conditions of the previous section : 



. f^i/ /-fry \ . E 2 / / \ . 



e 2 = —i Q < — 1 — ~ — ~ ) sin©i£ -\ { t r sin coot 



°l©i\ f+g + h) (oAf+g + IJ 



, E 8 / <y \ . 1 



Caw II. 



In the case in which all three circuits are identical and 

 undamped and the coupling coefficients are all equal, 

 equation (9) reduces to 



{«»(! + «)-*»}*{»»(! - 2a) -P} =0, 



> (19) 



