Triple Pendulums ivith Mutual Interaction. 



21 



Thus, by substituting in the first and second lines of (IS) 

 the values of oc and y from (17), we find 



(/_ m 2)Ee mit - kFe mit = Jcz, j 

 (h - m 2 )Fe mit ~ kEe mii = lz> 1 ' 



Eliminating z between these two, we obtain 

 fl — lm 2 +k 2 , 



(24) 



(25) 



Again, from the first line of (18) using (17) and (25), 

 we find 



\~T~k(h-m ! ' + T)J- Le ~^ e ' 



which reduces to 



G = 



k{h-m 2 + l) 



E = y(E), say. . (26) 



It may be seen from (25) and (26) that the functions 

 <f> and yjr have different values for m 1? m 2 , and m 3 , but are 

 alike for the positive and negative signs of any given 

 numerical values of m. Thus the-functions may be written 

 <£ 1? (f> 2 , and c£ 3 for +mj, +m 2 , and ±m 3 respectively. The 

 same applies to the aJt's. 



(3) Phases.— We may now use (23), (25), and (26) in 

 (17) and write the general solution as follows : 



x = E x e m ^ + E^e~ m ^ + E 2 e m ^ -f EJer™* 



+ 2 terms in _Z£ 3 , 

 y = ^(EJe™^ + bfJEAe-™* + ^ 2 (E 2 )e m ^ 



+ (j> 2 (E 2 y~ m * it + 2 terms in (/) 3 (£V), 

 * == ^(jEi)^* + ^(JEiO*"* 1 * + f 2 (E 2 )e m '^ 



+ ^ 2 (4}'>~ ma * + 2 terras in ^ 3 C#a). j 



(27) 



Owing to the fact that ^ and 1^ do not alter with the 

 algebraic sign of on, these may be written in the more 

 compact form given below in which the phase angles are 

 constant in any one column 



