Small Vibratory Motions of Elastic Fluids. 305 



dr- ^ \dr ^ rdr) 4r«' 



•• '^^•J¥ ° (^<= ~ dr' "^ 4rr 



This equation is integrable whenever ^ may be neglected, that 



is, when r is not small, because is a very small quantity. 

 The integral is, 



<p^= F,{r-at) +l(r + at); 

 ••• as = -l^^^ = -^^{Fir-at)-fir + at)], 



and V = ^/»- + </" = -r- . 

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



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



as % has been before neglected. The interpretation of this in- 

 tegral will be understood by what will be said of the integral of 



df ~ \dx- df "^ rfxV' 



which applies to motion in space of three dimensions. To obtain 

 the latter integral, put x'' + y" + z- = r". A process like the pre- 

 ceding will lead to the exact equation 



r0 = FXr-at) +f,{r + at) (B). 



... d(b d<b d<b dtp 



A\soa^s + -^ = 0, p=^, 5, = ^, u = -^; 



Vol. HI. Part I. Qq 



