38 



DYNAMICAL THEORY OF SOUND 



This is a cubic in n\ One root is nf = 2/z, and we find on 

 reference to (10) that this makes B l = 0, A l = - C 1} and there- 

 fore 



x = A l cos (njt 4- 6j), i/=0, z- A 1 cos(n l t + e l ). ...(12) 

 This mode might have been foreseen, and its frequency 

 determined at once, as in the preceding example. The 

 remaining roots of (11) are 



and it appears from (10) that these make 



A 2 =C 2) B 2 = -</2A 2 , and A 3 =C 3 , B 3 

 respectively. The corresponding modes are therefore 

 x = A 2 cos (n + 6 2 ), y = - V2 A 2 cos (n 2 t + e 2 ), 



z A 2 cos (n 2 t + 6 2 ), 

 and 



x A 3 cos (n s t + e 3 ), y = V2 A 3 cos (n 3 t + e 3 ), 



^ = J. 3 cos (n 3 ^ + e 3 ). 

 These are shewn, along with the former mode, in Fig. 18. 

 The complete solution of the equations is obtained by super- 

 position of (12), (13) and (14), and contains the six arbitrary 

 constants A lf A 2 , A S) e l} e 2 , e 3 . 



.(13) 



.(14) 



Fig. 18. 



We conclude these illustrations with the case of the double 

 pendulum, where we are entirely dependent on general method. 

 A mass M hangs from a fixed point by a string of length a, 

 and a second mass ra hangs from M by a string of length b. 

 For simplicity we suppose the motion confined to one vertical 



