291 Lord Rayleigh on the Free Vibrations of 



If there were no rotatory terms, the above system of 

 equations would be satisfied by supposing one coordinate <p r to 

 vary suitably, while the remaining coordinates vanish. la 

 the actual case there will be in general a corresponding- 

 solution in which the value of any other coordinate (f> s will 

 be small relatively to <f> r . 



Hence if we omit the terms of the second order in j3 } the 

 rth equation becomes 



(c r -* r *a r )<l> r = 0, (5) 



from which we see that a r is approximately the same as if 

 there were no rotatory terms. 

 From the sth equation we obtain 



terms of the second order being omitted ; whence 



i<r r /3 sr _ i<ifi*r ,„. 



*•'*-— ^^--.w-O' • • ' (6) 



where on the right the values of a n <r s from the first approxi- 

 mation (5) may be used. This equation determines the 

 altered type of vibration ; and we see that the coordinates 

 <p s are in the same phase, but that this phase differs by a 

 quarter period from the phase of cj> r . 



We have seen that when the rotatory terms are small, the 

 value of a r may be calculatod approximately without allowance 

 for the change of type ; but by means of (6) we may obtain 

 a still closer approximation, in which the squares of the /3's 

 are retained. The rth. equation (4) gives 



r + 2l a („*_„* ( 7 ) 



^ =<S ' ' -«>/-*/ 



Since the squares of the o-'s are positive, as well as a r , a si c r , 

 we recognize that the effect of /3 r s is to increase a r 2 if a- r 2 be 

 already greater than as 2 , and to diminish it if it be already 

 the smaller. Under the influence of the /3's the cr's may be 

 considered to repel one another. If the smallest value of <r r be 

 finite, it will be lowered by the action of the rotatory terms*. 

 The vigour of the repulsion increases as the difference 

 between <r r and a s diminishes. If cr r and a s are equal, the 

 formulae (6), (7) break down, unless indeed j3 rs = 0. It is 



* This conclusion was given in Phil. Mag. v. p. 138 (1908), but without 

 some reservations presently to be discussed. Similar reservations are 

 called for in ' Theory of Sound/ §§ 90, 102. 



