46 Oscillations in Three Coupled Electric Circuits. 



thus giving as the three roots 



ft>2 2 = 



0) 3 2 = 



z 



(20) 



1-2* 



It will thus he seen that in this case there are only two 

 periods of oscillation proper to the coupled system. 



Case III. 



As a check on the theory, let us now consider the case in 

 which the couplings are put equal to zero, and also, to 

 further simplify matters, let the circuits be undamped. 



Then equation (9) gives the following values for the 

 frequencies : 



»!»=/*, a> 2 2 = m\ o) 3 2 = n 2 , . . . (21) 

 and equations (18) give 



ai--, 



a 2 = a 3 = 0, E = F = 0. 



Thus the case reduces to a single oscillation in the first 

 circuit given by 



e x = -j- sin It, ...... . (22) 



as was to be expected, since the circuits are now entirely 

 separate. 



Summary. 



In the present paper the mathematical theory of the 

 ^oscillations in three coupled electric circuits is developed 

 by a simple method. It is shown that the problem involves 

 the solution of a certain, cubic equation involving the 

 squares of the frequencies of oscillation proper to the 

 system. This equation is obtained, and is then solved for 

 certain special cases. 



The author desires to express his thanks to Prof. E. 

 H. Barton, F.R.S., for his ever kindly interest and useful 

 advice in connexion with the above work. 



Physics Department, 



University College, Nottingham, 

 Feb. 3, 1921. 



