Small Vibratory Motions of Elastic Fluids. 307 



in which the fluid was disturbed, in the investigation of the 

 equations of the motion, and because the origins of r and t are 

 quite arbitrary. Also because v and as will have the same values 

 for a given value of r and t, in whatever direction r be drawn from 

 the origin, supposing the forms of the arbitrary functions to be 

 the same for all directions, it follows that the general character 

 of the motion is spherical ; that each particle may be considered 

 as moving in the direction of the radius of a sphere, and its 

 motion to be some function of that radius. 



15. As in obtaining the equations 



v = ^{F{r-at)+f{r + at)\, 



as=^{F{r-at)-f{r + at)\, 



for the motion in space of two dimensions, and the equations 



v = ]^{F{r-at)+f{r + at)\, 



as=~^{F{r-at)-f{r + at)\, 



for the motion in space of three dimensions, only terms of the 

 oi'der % were neglected, it follows that these equations are appli- 

 cable to most cases that can occur; for the above general character 

 of the motion shews that-^ will almost always be a very small 



quantity. We shall confine our attention at present to the latter 

 equations. By reasoning upon them exactly as we reasoned on 

 the analogous equations in rectilinear propagation, the following 

 deductions will be made : — 



1st. The velocity of propagation is always equal to a. 



QQ2 



