THEORY OF WAVES OF FINITE LONGITUDINAL DISTURBANCE. 
281 
probably in many solids, the quantity of heat received during an indefinitely small 
change of pressure dp and of bulkiness ds is capable of being expressed in either of the 
following forms : 
7Y/<p 
T 
=c. 
dr 
dp 
dr 
d l> + C Pds ds; 
in which c s and c p denote the specific heat at constant bulkiness and at constant pres- 
dr 
sure respectively ; and the differential coefficients ^ and of the absolute temperature 
are taken, the former on the supposition that the bulkiness is constant, and the latter 
on the supposition that the pressure is constant. Let it now be supposed that the 
bulkiness varies with the pressure according to some definite law ; and let the actual 
ds 
rate of variation of the bulkiness with the pressure be denoted by Then equation 
(12) may be expressed in the following form : 
dr . dr ds 
dp Cp ds ' dp 
= 0 . 
Now, according to the dynamic condition of permanence of type, we have by equa- 
tion (6), 
ds 1 
dp rri 2 ’ 
which, being substituted in the preceding integrals, gives the following equations from 
which to deduce the square of the mass-velocity : 
£^.{mx|-c p |}=0. (13) 
dT 
It is sometimes convenient to substitute for c p -j the following value, which is a known 
consequence of the laws of thermodynamics : 
dr dr . r dp 
Cp -= C S -+T-£> 
ds 
J dr 
(13 a) 
dj) 
the differential coefficient being taken on the supposition that s is constant. The 
equations (13) and (13 a) are applicable to all fluids, and probably to many solids also, 
especially those which are isotropic. 
The determination of the squared mass-velocity, to 2 , enables the bulkiness s for any 
given pressure q>, and the corresponding velocity of disturbance u, to be found by means 
of the following formulae, which are substantially identical with equations (8) and (3) 
respectively : 
5 = s +^?; (U) 
MDCCCLXX. 
u=7n(S—-s )=^- — - 
m 
2 p 
(15) 
