OF THE MOTION OF FLUIDS. 199 



instant the velocity and density at any given point, which result from 

 a given disturbance. In other cases in which the velocity of propaga- 

 tion is variable, the determination would be more difficult, but must 

 probably be arrived at by the same principle of reasoning. Variable 

 propagation is analogous to variable motion, as uniform propagation to 

 uniform motion, and would seem to require integration to determine 

 the time at which the effect of a given disturbance is felt at a given 

 place. 



16. If in the equation (C), a be an indefinitely great quantity, 

 the terms which do not contain a^ as a factor may be neglected in com- 

 'parison of those which do, and the equation will become 



dr^ ^ dr\r ^ r + l) ' 



which by integration gives -^ = — j-, the same as for incompressible 



fluids. This result was to be expected, because a, as is well known, 

 is the velocity of propagation in the compressible fluid, and when this 

 becomes infinite, the propagation is instantaneous, and the fluid there- 

 fore incompressible. 



If / be indefinitely great, it will be found in the same way that 



-r~ — — , and the motion is such as was considered Art. 3. 

 dr r 



Let now -^ be very small compared to «, and X, V, Z, and / 

 each = 0. The equation (C) reduces itself to 



"-11?^^"^ dr df-^' '''''• dr' -~dF~' 



a particular integral of which is r^=^'P{r — at). This gives 



d^ ^ F\r-at) _ F{r-at) 



dr ~ r r^ ^ "' 



CC2 



