Prof. Challis on the Nature of Adrial Vibrations. 465 

 Now, as is known, for plane-waves 

 diso _ dp 

 dz ~~ * pdz' 



Hence the condition of uniform propagation of a given state 

 of density is not satisfied in this case of motion. Next let us 

 take the case of ray-vibrations. We will suppose the axis of 

 z to be the axis of the vibrations, so that « = and v=0. 

 Then equation (3.) becomes 



du dv dw , x dp 



which, since 



df df J<p r ' , d?f d?f & 



r dx r dy' J dz J da? dy* d? 



gives 



b* d*<p _ ( d<p\ dp 



<P o* + d^-\ ai ~Tz)p~dz'' 



Differentiating now the equation (2.) along the line of mo- 

 tion, we have 



a*dp d*<p dtp d?<p 

 pdz df 2 dz dz* ■ 



Hence, substituting in the above equation, 



-i** + a« ft + (a - &\ ( ft + * **\ -0 



0(p + a W + V' Tz)\dzlt + Tz 1^)-°' 



Consequently, by comparing with the equation which I have 

 called (B.), viz. 



-Wa-l.rfft~.ft -2^. ft _ a< ? ft-n 



° 9+a dT* dF dzd^dt l&'dF- ' 

 the result is, 



dt* +a ' dzdt + dzV 1 dz* + dzdt)-°- 



Putting 3> for -^ +a x -^, and-^ for -X this equation be- 

 comes 



d® d^ dz 



~dJ + Wdt > 

 or 



