by means of a Double Pendulum. 



571 



A A 



The ratios ~ and ^ can be found most simply from 



-Dj J3 2 



equation (2) with the respective values of n 2 inserted. Thus 

 we have 



__ *hf A r9 



15 1 — 1^ /YJil 2\„.2 • ^1' * •■ • * W 



B, = 



ah 2 g 2 



A 2 . 



(10) 



#A 2 -(K2 a + /*2 2 ),f 



To obtain the special solution for any particular case, 

 we must know the initial displacements and velocities 

 and determine the four constants A, B, a, and ft. As a 

 preliminary to this, we can write 



6 — A l cos (pt + u) + A 2 cos (qt + /3). 



ah 2 p 2 



4> = 



ali-2 q 2 



+ 



• A x cos (^ + «) 

 2 . A 2 cos (gtf 4- ft), 



9 h 2 -(K 2 2 + hJ)q 

 6 = — A x p sin Q»£ + a) — k 2 q sin (<^ + /3) , 

 ah 2 p 2 



$ = 



gh 2 -(K 2 2 + h 2 2 )p 

 ali 2 q 2 



gh 2 -{K 2 2 + h 2 2 )q 



2 . Aip sin (pt+ a) 

 2- A 2 #sin {qt + ft), 



(id 



(12) 

 (13) 

 (14) 



From these equations we can obtain the particular solution 

 in any case from a knowledge of the initial conditions. 



The values of y and — the ratio of the two superposed 



S.H. vibrations; and the relation between (i.) y and a. 

 (ratio of the distance from the point of suspension of A to 

 that of B) to the whole length of A, or rather the length of 



simple pendulum equivalent to A, and (ii.) 7 and -, is 



represented in graphs (i.) and (ii.). 



Graph (iii.) is given to show the relation between 7 



and - in the electrical problem similar to the mechanical 



9- . 



one in question. 



An examination of graphs (ii.) and (iii.) and the photo- 

 graphic reproductions given in Plate XI. shows how in 

 both cases (mechanical and electrical) the difference between 

 the two superposed S.H.M.'s becomes greater and greater 

 with the progressive increase of coupling. 



