Small Vibratory Motions of Elastic Fluids. 276 



of indefinite extent, and to be unsolicited by any extraneous 

 forces. Let v = the velocity of 'the particles at the distance a 

 from a fixed origin, and at a time t reckoned from a given 

 epoch ; s = the condensation at the same distance, the mean density 

 of the medium being = i ; and a" a constant proportional to its 

 mean elastic force. The usual investigation leads to the equa- 

 tions, 



f^=«'S w. 



g+«--° <»). 



v.^i (3). 



ax ^ 



The integral of equation (i) is 



<{, = F,(x - at) +f,(x + at). 



Hence v = F (x - at) +f{x + at) (a), 



as = F[x - at) -f{x + at) (/3).. 



It is particularly to be observed with respect to these equations, 

 that the origins of x and t are perfectly arbitrary; and that as 

 the equations were investigated without reference to the manner 

 in which the particles were put in motion, all results derived 

 from them, prior to any hypothesis about the mode of dis- 

 turbance, must be quite general; that is, must obtain in what- 

 ever way the particles have been caused to move, provided 

 always that v be very small compared to a. Because each of 

 the functions F and /, satisfies separately the differential equa- 

 tions, the motion which results from the consideration of either 

 of them by itself, will be possible, though not the most general 

 that can obtain. Suppose we consider the function F. Then 

 V = as = F{x- at). These equations shew that in the case sup- 



M M 2 



