358 Lord Rayleigh on the Vibrations of 



quantities ; whence we obtain as the Lagrangian equations of 

 motion : — 



([l]D 2 + 8[l]D 2 +{l}+S{l}> 1 + (5[12]D 2 + a{12}> 2 



+ ...=0, 



(8[12]D 2 +5{12}> 1 + ([2]D 2 +8[2]D 2 +{2}+5{2})</) 2 



+ ...=0. 



In the original system the fundamental types of vibration are 

 those which correspond to the variation of but one coordinate at 

 a time. Let us fix our attention on one of them, involving, say, 

 a variation of <f) r , while all the remaining coordinates vanish. 

 The change in the system will in general carry with it an altera- 

 tion in the fundamental or normal types ; but under the circum- 

 stances contemplated the alteration is small. The new normal 

 type is expressed by the simultaneous and synchronous variation 

 of the other coordinates in addition to <j> r ; but the ratio of any 

 other coordinate <f> s to <f> r is small. The determination of these 

 ratios constitutes the solution of the question proposed. 



Since the whole motion is simple harmonic, we may suppose 

 that each coordinate varies as cos pt, and substitute in the dif- 

 ferential equations — p* for D 2 . In the sth equation <j> s occurs 

 with the finite coefficient, 



The coefficient of <f> r is — S[rs]jo^ + B{rs] . The other terms are 

 to be neglected, inasmuch as both the coordinate and its coeffi- 

 cient are small quantities of the first order. Hence 



9s *' W-rtM ( ' 



Now approximately 



and therefore 



, A= SM.#-8{rg} 



9s ' 9r ~WpfW~' 



the required result. 



If only the kinetic energy undergo variation, 



*.:*,= *LM; ...... (3) 



Pi-Pi W 



From the rth equation we see that — p*[r] —p 2 r &[f] + {r}-f67r} 

 is a small quantity of the second order; so that 



