) « ^«"7 



Triple Pendulums with Mutual Interaction. 21 



for small vibrations if isolated from the other pendulums. 

 The depth of any bob below the level AE of the ends of the 

 bridle is called a, and is clearly the virtual length for oscilla- 

 tions of all the pendulums in phase together. From analogy 

 of the coupled-pendulums dealt with in earlier papers (Phil. 

 Mag., Oct. 1917, Jan. 1918, July 1918) it was anticipated 

 that the frequencies involved would be functions of a, 6, 

 and c, and so these lengths were forced into the analysis as 

 early as possible. 



But, in addition to these three lengths, there are others 

 which must be temporarily used. Thus the depth of the 

 apex D of the bridle below AE is called d. Further, with b 

 is associated the length b (which equals a — b) and b', which 

 is the depth of B below AE, on figs. 1 and 2. Similarly, 

 c = a — c and c' is the depth of C below AE. 



For the symmetrical arrangement in the side elevation as 

 shown in fig. 2 (i.e., AD = DE) we find the following rela- 

 tions between the constants in question, to which are added 

 those just mentioned : 



b = a — b, c = a — c, "^ 



,_ 2db _ 2d(a-b ) ,__ 2d(a-c) 



~ d + bo ~ d-Ya— b ' d+a—c' 



d h , = d{d-a + b) d c , = d(d-a4-c) ^ 

 dj-a — b ' d + a — c 



!'-&' = 



2d*(b-c) 



(d + a — b)(d + a — c) 



(i). 



(b) Equations of Motion and their Reduction. 



The theory now to be developed is based upon the following 

 assumptions and data : — 



(i.) The oscillations contemplated are to be regarded as 

 small. 



(ii.) The damping of the oscillations by friction, etc., is to 

 be considered negligible. 



(iii.) Total lengths of all bridles and suspensions to be the 

 same. 



(iv.) Masses of bridges and cords all negligible. 



(v.) Bridges to be considered rigid. 



(vi.) Changes in masses of bobs by discharge of sand to 

 be neglected. 



(vii.) Masses of P, Q, and R to be equal (each of value 

 2M). 



