PAPER BY LORD RAYLEIGH. 293 



When n is small we have approximately 



", ^ > (21) 



' (J 

 so that p vanishes when 



e'''=^.„ {2'J) 



n-a- 

 or by (i) wheu 



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

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



(7 p dz, vanishes, as of course it should do. 

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



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



^^ ' 3(iUU 



We takea=33xl0^^=981. Then 



o-o~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=(), and that u, u\ p, are functions of 

 X and s only. Further we may suppose that x occurs only in the factor 

 e''" ; that is, that the motion is periodical with respect to x in the wave- 



27T 



length -x~; and that as before t occurs only in the factor e"" . Equa- 

 tions (5) and (6) then become 



dw gic ,, ,,,, 



t«p+tA-M-f-^---^=0 . . . . (24) 



d^ k p = — mi (2o) 



dz - ^ ' 

 from which if we eliminate u and w we get 



^-|/i+($-^0 "=» • • • • <^^> 



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



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

 the equations are satisfied by 



7i'=}c'a' (28) 



