Oscillatio?is in Three Coupled Electric Circuits. 41 

 solution of equations (2) may be 



Then the genera 

 written in the form : 



a 3L =a 1 e-e'cos (co } t + d x ) + a 2 e~ rt cos (w 2 t + 6 2 )-{-a s e- st cos (co s t + O s ) , 

 e 2 = bie~ qt cos {a>it + fa) -\-b 2 e~ rt cos (&> 2 £ + (£ 2 ) + b z e~ 8t cos (<*>3t-\-fa), 

 e 3 = c 1 e~ qt cos (go ±t + \jr { ) + c 2 e~ rf cos {co 2 t-\- ^2) + c 3 <?~^cos (o) 3 £ + i/r 3 ), 



(ii) 



ix = a { e~ qt {(j cos (co^H-^) -f- ©i sin (©^ -f^)} ~"i 



+ « 2 e~ r/ {r cos (g> 2 £ + #2) + w 2 si' 1 (^ + #2)} 

 + a 3 <?~ s *{s cos (co 3 t -f 3 ) -f w 3 sin (o> 8 £ + 3 ) }, 



i 2 = l^e-^lq cos (corf -r-<^) 1 ) + ft) 1 sin (tM + ^i)} 



+ ^-^{?'Cos(&) 2 ^i-</>2) + &) 2 sin (&) 2 ^ + <^ 2 )} }> (12) 



4 btf-*{s cos (o> 3 £ + fa) + w 3 sin (© 3 * + fa)}, 



h = c 1 g-^{gcos(&) 1 ^ + ^ 1 ) + a) 1 sin (cM + ^i)} 



+ c 2 e~ rt {r cos (w 2 £ + ^2) + ^2 sin (g> 2 £ + ^2) I - 

 -f c 3 e~ s *{s cos (0)3^ + ^3) + a> 3 sin (g> 3 £ + ^ 3 )}, 



in which a, &, c, #, <£, i|r are arbitrary constants and the 

 values of the other quantities are given by (9) and (10). 



From equations (11) and (12) we see that in each of the 

 coupled circuits the electromotive force and the current 

 each consist of three damped harmonic oscillations, of 

 which the dampings and periods can be calculated as 

 above in terms of the couplings and the constants of the 

 separate circuits. We also see that the same three oscil- 

 lations occur in each of the three circuits, only the 

 amplitudes and phase-angles being different. 



An attempt to visualize the resultant of three simul- 

 taneous harmonic oscillations presents a more difficult 

 problem than the familiar case of two simultaneous 

 harmonic oscillations. Any means by which the signi- 

 ficance of equations such as (11) and (12) can be grasped 

 immediately without having recourse to the somewhat 

 tedious process of drawing graphs of the oscillations under 

 various conditions would be welcome. Reference may 

 therefore be made to the recent work of Barton and 

 Browning*, "Triple Pendulums with Mutual Interaction 

 and the Analogous Electric Circuits. — I.," in which the 

 traces of actual oscillations of the analogous case of three 

 mechanical vibrators are exhibited. These bring out in 

 a very clear manner the somewhat unexpected nature ot 

 the resultant vibration. 



* Phil. Mao-. Nov. L920. 



