﻿by Vibrations of the Air, 281 



(d<p) for this differential, we have 



(d(j)) dec dy dz 



ds ds ds ds 



u v w; 



Hence 



dY_d^ CdV _d$ f 



dt ~ ds dt* J dt ~~ dt. 



By integrating the equation (a), 



a«Nap.logp--f J&- J+/feft y , *), 



a, /3, 7 being the coordinates of a given point on the line s. 

 Now, if we suppose that the mass of fluid is of unlimited dimen- 

 sions, and that at a certain position it is agitated by the vibra- 

 tions of a small body, on tracing from the source of disturbance 

 the course, at any given instant, of any line s coinciding through- 

 out in direction with that of the motion, a position will be reached 

 at which the motion is of insensible magnitude ; and this will be 

 the case even if the course of s should be modified by incidence 

 of the vibrations on a small solid obstacle. At every such posi- 



dV 

 tion -=- =0, V=0, and the density is that of the fluid at rest. 



Calling this density -or, we shall have 



a' 2 Nap.log^=/(*,/3, r ,0, 

 and 



V 2 



a' 2 N ap.log£ = -J^- 



Hence we obtain the exact equation 



n 1 CdV, V2 



(2 \ ( 



consequently, by expanding to terms of the second order, and 

 putting -57(1 + 0-) for p> 



'dV , V 2 1 / CtiY . \ 2 



di ds ~ Y + wA)-dt ds )' 



Now the only part of the pressure a ,2 <r which can give rise to a 

 motion of translation is expressed by the non-periodic terms of 

 the right-hand side of this equation. Hence, omitting the term 



JdY 

 ~jr ds, which, as we have seen, is wholly periodic whether or 



not, it contains small quantities of the second order, and putting 



