178 Lord Rayleigh on the 



In discarding the restriction to one dimension, we may 

 suppose, without real loss of generality, that v = 0, and that 

 u, iv, p are functions of x and z only. Further, we may 

 suppose that x occurs only in the factor e ikx ; that is, that the 

 motion is periodical with respect to x in the wave-length 

 2ir/k ; and that as before t occurs only in the factor e int . 

 Equations (5) , (6) then become 



., div aw „ /C%AS 



inp+zhi+^-^ = 0, .... (24) 



a?kp=—nu, (25) 



a*dpldz=—inw; (26) 



from which if we eliminate u, w 3 we get 



s-rJH^>=°> • • • ™ 



an equation which may be solved in the same form as (12). 



One obvious solution of (27) is of importance. If dpjdz = 0, 

 so that w=0, the equations are satisfied by 



n«=AW (28) 



Every horizontal stratum moves alike, and the proportional 

 variation of density (p) is the same at all levels. The possi- 

 bility of such a motion is evident beforehand, since on account 

 of the assumption of Boyle's law the velocity of sound is the 

 same throughout. 



In the application to meteorology, the shortness of the 

 more important periods of the vertical motion suggests that 

 an " equilibrium theory " of this motion may be adequate. 

 For vibrations like those of (28) there is no difficulty in taking 

 account of the earth's curvature. For the motion is that of 

 a simple spherical sheet of air, considered in my book upon 

 the ' Theory of Sound,' § 333. If r be the radius of the 

 earth, the equation determining the frequency of the vibration 

 corresponding to the harmonic of order h is 



nV=A(A+l)a 2 , (29) 



the actual frequency being n\2ir. If r be the period, we 

 have 



r- f" y » (30) 



a\/h{/i + l) V ; 



For h = l, corresponding to a swaying of the atmosphere 



