452 Lord Rayleigh on the Finite Vibrations of a 



where 



V 3 = 7i^i 3 + 72^>i 2 </>2 + 73</>i^2 2 + 74^2 3 , • • (3) 



giving as Lagrange's equations 



a, d 2 fa/dt 2 + c x fa + Sy x fa 2 + 272^^ + 7s tf>2 2 =0, . (4) 



a 2 d 2 fa/dt 2 + c 2 fa + 37 4 fa 2 + 273 fa fa-\-y 2 fa 2 = 0. . (5) 



To satisfy these equations we assume 



^> 1 = H + H 1 cos7?£ + H 2 cos2?i£ + H 3 cos 3nt + . . . . (6) 



fa — K + K x cos nt + K 2 cos2^ + K 3 cos3n^ + . . . . (7) 



In general we may take as one approximate solution 



<j) i = 'E l cos nt, fa = (8) 



with 



^ = ^1; (9) 



and in proceeding to a second approximation we may regard 

 all the other coefficients as small relatively to H4. On this 

 supposition the 4th and 5th terms in (4) may be omitted, 

 so that fa is separated from fa. Substituting from (6) and 

 equating the terms containing the various multiples of ?it y 

 we get 



ox H + §7^=0, 



(c 1 -4/i 2 « 1 )H 2 + |7iHi 2 = 0; 



_ 371 Hi 2 , TT ™„, 3 7lH! 



with 



c x = n 2 a u 



as in the first approximation. In like manner 



c 2 K + i7 2 H 1 2 = 0, 

 (c 2 — n 2 a 2 )K 1 = 0, 

 (, 2 _4n 2 a 2 )K 2 + 172^ = 0. 

 Thus, if c 2 differs both from n 2 a 2 and from 4?r« 2; we have 



so that 



2 



fa= y -±— — + H.! cos nt n . } cos 2nt, . (10) 



