THEORY OF VIBRATIONS 



37 



deflections of the two particles by x, y, the superposition of 

 the two modes gives 



x = A cos (nj + eO + B cos (n + e 2 ),) 



\ (b) 



y = A cos (%< 4- 61) B cos (n z t -4- e 2 ),| 



where the four constants JL, 5, e 1} e 2 are arbitrary. 



In the case of three attached particles the nature of the 

 various modes is not so immediately obvious, even in the case 

 of symmetry. We will suppose that the masses are equal, 

 and that they divide the line into four equal segments a. 

 Denoting the deflections by x, y, z, we have 



-P 



dt~ a a ' ( 



dt 2 a, a ) 



If we put, for shortness, fi = P/Ma, these may be written 



.(7) 



.(8) 



To ascertain the existence of modes of vibration in which 

 the motion of each particle is simple-harmonic, with the same 

 period and phase, we assume, tentatively, 



x = A cos (nt -f e), y = B cos (nt + e), z C cos (nt -f e). (9) 



It appears, on substitution in (8), that the equations will be 

 satisfied provided 



(10) 



These three equations determine the two ratios A : B : C 

 and the value of ri 2 . Eliminating the former ratios we have 



............. (11) 



