

MECHANICAL ACTION OF HEAT. 163 
Fourthly, Let the body be compressed, without change of temperature, to its 
original volume V; then the heat given out is 
TtT—K f[1 aU 
+8V oom (y-zv) 

while the power absorbed in compression is 
—dV.P 
The body being now restored in all respects to its primitive state, the sum of 
the two portions of power connected with change of volume, must, in virtue of 
the principle of vis eva, be equal to the sum of the four quantities of heat with 
their signs reversed. Those additions being made, and the sums divided by the 
common factor 6 V 67, the following equation is obtained :— 
qdP_ 1 1 dU 
drt CaM +-7¥) 
The integral of this partial differential equation is 

(9.) 
1 dP 
U=¢.7+ fav y—coaM5-) lve (1D 
Now ¢ . 7 being the same for all densities, is the value of U for the perfectly 
aseous state, or K, for in that state, the integral = 0. 
g = gr 
The values of the partial differential coefficients are accordingly— 
au 1 dP 
ave NCRMG; 
dU K aP (11) 
— Cam fav ae 
and they can, therefore, be determined in all cases in which the quantity 
k=C nb, and the law of variation of the total elasticity with the volume and 
temperature are known, so as to complete the data required in order to apply 
equation 6 of this section to the calculation of the mechanical value of the varia- 
tions of heat due to changes of volume and molecular arrangement. 
The total elasticity of an imperfect gas, according to Equations VI. and XII. 
of the introduction, being 
P= 
T 
ep 
omy (I-F (0.2) ) + f(D) 
its first and second partial differential coefficients with respect to the tempera- 
ture are,— 
dP 1 d x 
7 a conv (2 a (147 a 4 (Dz)) 
d?P 1 d d? T 
a =- amv (23 + " 773) F (0.2) 

VOL. XX. PART I. 
