Triple Pendulums with Mutual Interaction. i>i:' 



Substituting these in (1),(2), and (3), we obtain 



p*M. 2 x+p*M 1 y + Q^ ~L 3/ >^=0. 



(6) 



% Eliminating #, y, and z from (G) by the method of deter- 

 minants, we have 



This gives an equation of the sixth degree in p, viz.: — 

 / [2M,M 2 M 3 - L^L, + MI, 2 + L. 2 M 2 2 + L 3 M 3 2 ] 

 t ryU-Mi 2 , L,L, -M/ , L,L 2 -M 3 2 1 

 +i ' L 8, S 2 + S 3 J 



_ -rtk + m + m\ + ms s =°-- ■ ■ (8) 



This is a cnbic equation in p 2 , of which the roots may be 

 written, p 2 =p i 2 , p 2 2 , Pn 2 . 



Then p= ±p { , ±p 2} ±p z ; 



but the negative signs may be disregarded as they introduce 

 nothing new. 



Hence vibrations of three periods are set up. 



Thus the general solution of the equations may be written 



a?=E 1 sin (p l t + e 1 ) + F 1 sin (p 2 t + </>{) + G, sin (?V + %i), ( 9 ) 



and similar equations for y and z. 



If the circuits were vibrating each isolated from the others, 

 then their vibrations would be proportional to 



sin It, sin mt, and sin nt, . . . (10) 



where /2 1 ^ 



Jj 2 02 

 1 



" L3S/ ) 



(H> 



