Vibrations elucidated by Simple Experiments. 259 



equal to the weights of the bobs simply. The equations of 

 motion may accordingly be written as follows : — 



P J+P^=0 and 



d 2 z 



Q§ + (^^>=^- • w 



The coefficient of coupling 7 is given by 



^ = (P a + Q)*W (57) 



In the special case where P = Q, and r = s, which will 



henceforth be adhered to, if we write m 2 for gjr, the above 



become 



d 2 y 



-~ + m 2 y = m 2 a.z, (58) 



d 2 " 



jl + (l + u + u 2 )m 2 z = m 2 « h . . . (59) 



a. 



and 7 2=— — — (60) 



' 1 + a + ar x ' 



So for very small values of a, y — a nearly. 



Solution and Frequencies for Equal Bobs and Lengths. — 

 To solve these equations try in (58) 



y = e xt (61) 



This gives 



2= ^+m 2 ^ (62) 



Then (61) and (62) in (59) give the auxiliary equation in #, 

 x i + x 2 (2 + <x + u 2 )m 2 +{l + ot)m 4 = 0. . . (63) 

 If we write this in the form 



x i + a 2 ( 1 ) 2 + q 2 )+p 2 q 2 = 0, .... (64) 



we see that 



x=±pi or ±.qi (65) 



