PAPER BY LORD RAYLEIGH. 293 



When n is small we have approximately 



5 9 



m 2 = - 

 



so that p vanishes when 



gz 



or by (4) when 



n 2 d 2 





(23) 



Below the point determined by (23) the variation of density is of one 

 sign and above it of the contrary sigu. The integrated variation of 



,.CO 



density, represented by | a p dz, vanishes, as of course it should do. 

 « J i> 



It may be of interest to give a numerical example of (23). Let us 



In 

 suppose that the period is one hour, so that in c. G. s. measure n= -r-^-r-- 



3600 



We take «=33x 10 4 , #=981. Then 



<r 290' 



showing that even for this moderate period the change of sign does 

 not occur until a high degree of rarefaction is reached. 



In discarding the restriction to one dimension, we may suppose, with- 

 out real loss of generality, that v=0, and that u, w, 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 -jr- ; and that as before t occurs only in the factor e int . Equa- 

 tions (5) and (6) then become 



dw giv . ._.. 



mp+iku+-fa— ^r=0 • • • • (* 4 ) 



a 2 h p=—nu (25) 



a?^/-=-inw (26) 



dz 



from which if we eliminate u and w we get 



t£-!4K^») '=» • • • • < 37 > 



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



One obvious solution of (27) is of importance. If ^ =0,so thatw=0, 



the equations are satisfied by 



tf^tfd 2 (28) 



