Triple Pendulums icith Mutual Interaction. 



25 



These three equations may, be reduced to the compact 

 form : 



^ +hy = lz+kx, 



dt 2 



cPz 



dt 2 



where 



iy = LZ-f-KX, j> 

 + hz = kx + ly, I 



(14) 



A = 



/__ #(3a 2 — 2a6 — 6c — 3ca) , 

 a(4a6— 36c — ca) 



,?(2a 2 6-6c 2 -c 2 a) 

 ca (4a6 — 36c — ca) ' 



,_ g(a — b)(a — c) 

 a(4a6 — 36c— ca) ' 



/ _ #( a ~ t')(2a6 — 6c — ca) 



ca(iab — 'dbc — ca) ' - 



(15) 



By analogy with the case of electric circuits (Phil. Mag. 

 p. 612, Nov. 1920) we might write for the three couplings 

 involved the symbols X, ii, and v. Then referring to (14) 

 and (15), we have 



\ 2 = P/A 2 , 



,2_ 7,2 



v 2 = k 2 /fh) 

 Or, by (15) in (16) we may write 



(16) 



\ = 



(a — c) (2ab — be — ca) 



~) 



2a?b-bc i 





c(a — b) 2 (a — c)' 



[ (16 a) 



(2a 2 b - be 2 - c 2 a) {2d 1 -f 2a6 - 6c - 3ca) j 



(c) General Solution. 



We have now to seek a general solution of the three 

 equations of motion summarized in (14) and (15). This 

 solution will be found to involve the frequencies, phases, 

 and certain general relations between the amplitudes of the 

 various vibrational components. It will be well to take 

 these three topics in the following order. 



