296 Vibrations of Systems affected ivitli Small Rotatory Terms. 



The disturbance of the frequency from its original value is 

 now of Wig first order in /3 12 , and one frequency is raised and 

 the other depressed by the same amount. 



As regards the ratios in which fa, fa enter into the new 

 normal modes, we have from (9) 



£ = ^=^=±^W«i). • • • (W) 



fa *0oPl2 



From (14) we see that in the new normal vibrations the 

 two original coordinates are combined so as to be in quad- 

 rature with one another, and in such proportion that the 

 energies of the constituent motions are equal. 



The value of any other coordinate fa accompanying fa and 

 (f> 2 in vibration a- is obtained from the sth equation (4) . Thus, 

 squares of j3's being neglected, 



( C -A)^ + ^i + ^ 2 =0 3 . . . (15) 



in which, if we please, we may substitute for fa in terms of 

 fa from (14). 



For the second approximation to <r we get from (15) and 

 the two first equations (4) 



in which the summation extends to all values of s other than 

 1 and 2. In the coefficients of the second terms it is to be 

 observed that j3i 2 =— j8 2 i, and that j3i 5 /3 52 =/3 2s /3a ; so that 

 the determinant of the equations becomes 



j ^-^-s-fjv. l j c ^ 2 _ s _^p*_ X -o-% 2 *=o- 



[ c s — a 2 a s \ { z c s — a 2 a s \ ' u > 



. . . (16) 



terms of the fourth order in f3 being omitted. In (16) 

 Cx — a 9 ^, c 2 — o- 2 a-2 are each of the order j3. Correct to the 

 third order we obtain with the use of (12) 



/„. 2 „ 2\2 2 ^12 X °"oPl2 _i_ °0 Pl2 T? a lh>2s" + a2H\a _A C\ 7\ 



(<r -°° > - ff » ^ + (^)f ± K^)| S c s -^a.- -°- (17) 

 whence 



^2 _2_ 1 „ Pl2 1 iPl2 2 _2°"0 y? UlP2s 2 + a 2Hls 2 MQ x 



