298 Lord Rayleigh on the Vibrations of 



symmetry a positive and a negative n must influence the 

 frequency alike, we conclude that (4) still holds so long as n 2 

 is neglected. Thus to our order of approximation the fre- 

 quency is uninfluenced by the rotation, and the problem is 

 reduced to finding the effect of the rotation upon that mode of 

 vibration to which (3) is assumed to be a first approximation. 

 The equation for £ is at the same time reduced to 



a? + .^» +r - - ■•.•••(*) 



Since v is itself of the order n, the first of equations (1) 

 shows that u, as well as f, satisfies (5). 



Taking u and v as given in (3) and the corresponding £ as 

 the first approximation, we acid terms u f , v', $', proportional 

 to rij whoss forms are to be determined from the equations 



iau'=—gd£'/da:,i' (6) 



i<rv / =—r/dg/dy-—2ncosx, . . . . (7) 



(ffild& + &ld!? + Y)(g 9 tt,J) = 0. . . . (8) 



They represent in fact a motion which would be possible in 

 the absence of rotation under forces parallel to v and pro- 

 portional to cos x. This consideration shows that u is an odd 

 function of both x and y, and v an even function. If we 

 assume 



M , =A 2 sin2a? + A 4 sin4a?4-..., ... (9) 



the boundary condition to be satisfied at x= +^ir is provided 

 Eor, whatever functions of y A 2 , A 4 , &c. may be. If we 

 eliminate $' from (6), (7), we find 



dv' dv! 2n . 

 a.r ay ia 



dA . _ dA± . , , , 2n . 



= — ^— sm Ix H — z — sin 4a? + . . . -J- — sin x ; 

 ay ay icr 



or, on integration, 



, 2ni dA . a dA . . dA± _ 



v = cos a* j-^ — icosZx—r^—icosla—Z— (10) 



<r dy dy 4 dy K ' 



dA /dy being the constant of integration . In (10) the A's 

 are to be so chosen that v' = when y= ±y- L for all values 

 of x between — ^tt and +i7r. 



From (8) we see that A 2 , A 4 , &c. are to be taken so as to 

 satisfy 



^_3A 2 =0, ^-15A 4 =0, &c, 



