of the Motion of Fluids, Sfc. 401 



this interval, if we omit the consideration of small quantities of 

 an order which may be neglected. It will follow that 



(o)' — a)) T = 2 — z' ; 



for o)' is greater or less than w, according as z is greater or less 

 than z. But as the densities must vary inversely as the dis- 

 tances between the centers of gravity of the equal small masses, 

 if p, p be the densities corresponding to z', z, respectively, 



- = -.. Hence 

 z p 



z p - p 



T p 



Now 01 -w is the variation of velocity at a given instant between 

 two points indefinitely near each other, and p -p \s a. like varia- 

 tion of the density. Therefore ^r =- — r-. As - is the ratio 



• as T pas T 



of a linear dimension of a given mass of the fluid, to the time 

 during which the state of density at one extremity of this line 

 passes into the state at the other extremity, and as it is inde- 

 pendent of the absolute velocity of the mass, it must be accurately 



the velocity of propagation. This quantity, -, in general will 



be a function of the abscissa and the time ; but if it be constant 

 and equal to a, 



fs^''-^' 0, = a hyp. log. ^ + 0(0; 

 and <p(t) =0, if o) = where p = i. 



This result accords with what is said above. It should be 

 observed that the preceding reasoning is altogether independent 

 of the constitution of the fluid, and that the equation 



is of very general application., 



3 F2 



