Systems affected with Small Rotatory Terms. 295 



clear that the original assumption that <j) s is small relatively 

 to (f> r fails in this case, and the reason is not far to seek. 

 When two normal modes have exactly the same frequency, 

 they may be combined in any proportions without alteration 

 of frequency, and the combination is as much entitled to be 

 considered normal as its constituents. But the smallest 

 alteration in the system will in general render the normal 

 modes determinate ; and there is no reason why the modes 

 thus determined should not differ finitely from those originally 

 chosen. 



A simple example is afforded by a circular membrane 

 vibrating so that one diameter is nodal. When all is sym- 

 metrical, any diameter may be chosen to be nodal ; but if a 

 small excentric load be attached, the nodal diameter must 

 either itself pass through the load or be perpendicular to the 

 diameter that does so (' Theory of Sound/ § 208) . Under the 

 influence of the load the two originally coincident frequencies 

 separate. 



In considering the modifications required when equal 

 frequencies occur, it may suffice to limit ourselves to the case 

 where two normal modes only have originally the same 

 frequency, and we will suppose that these are the first and 

 second. Accordingly, the coincidence being supposed to be 

 exact, 



c 1 /a 1 = c 2 /a 2 = a 2 (8) 



The relation between fa and fa and the altered frequencies 

 are to be obtained from the first two equations of (3), in which 

 the terms in fa, fa, &c. are at first neglected as being of the 

 second order of small quantities. Thus 



{c 1 — a 2 a ] )fa + iaft l2 fa = 0\^ . (9) 



(c 2 — a 2 a 2 ) fa — ia(3 [ ^^Oj' 

 in which the two admissible values of cr 2 are given by 



(c 1 -a l a 2 )(c 2 -a 2 v 2 )-v 2 P 12 2 = Q. . . . (10) 



If one of the factors of the first term, e. g. the second, be 

 finite, j3i2 2 may be neglected and a value of o 2 is found by 

 equating the first factor to zero ; but in the present case both 

 factors are small together. On writing <r for a in the small 

 term, (10) becomes 



(<T*-<?o 2 y- = eo'Pi2 2 K<h, (11) 



so that 



<r 2 -a 2 =±a ^ 12 / s /(a 1 a 2 ) > . . . (12) 

 or 



o-=<r ±i/3 12 /^/(a 1 a 2 ). . . . (13) 

 X 2 



