146 THE EEV. S. EAENSHAW ON THE MATHEIMATICAL THEOET OF SOTXD. 
the dynamical equation takes the form 
1 ■ / dy\ d% 
If now we assume 
(F'a)^=-^, 
^ ^ §0 
the integral of the dynamical equation will be 
|^= ipF'a . t—f’a^ 
with a =^5 and w=C+Fa = C+F^-yy 
40. These equations are true of any motion which can be generated by a piston 
moving subject to the laws of continuity. See art. 13. The last shows that the rela- 
tion between velocity and density is independent of the law of genesis of the motion. 
The medium may be, as a whole, in motion with the uniform velocity C+F(l), and the 
motion of the particles caused by the motion of a piston will be superimposed on this. 
For convenience we shall suppose the medium as a whole at rest, and .•. C+F(1) = 0. 
If there be a point, or any number of points, within that part of the medium which 
is in motion for which for all such points a=l, and the equation 
^=+F(l).«-/(l). 
Avhich is always true for all such points, shows that at those points x changes its value 
at the rate of F'(l) feet per second, ^. e. the front of the wave travels at the rate of 
feet per second. 
which is constant, and depends not at all on the laAV of genesis, but only on the assumed 
relation between pressure and density, and not on the general value of even that, but 
only on its limiting value when ^=fo- Now many ditferent forms of the fimction (p may 
give the same limiting value ; and consequently all the media corresponding to these 
various forms of p will transmit a wave, as a whole, Avith the same velocity. Hence if 
the relation betAveen pressure and density be given, the wave-velocity may be instantly 
deduced from the expression 
[ 
?0 
, or from its equal. 
VsJ.’ 
using the subscript 0 to signify that after the differentiation has been performed fo is to 
be written for f. 
41. Since w=C+F^j^, by differentiation Ave obtain 
du 1 /dp\^ 
