
DYNAMICAL THEORY OF HEAT. 479 
ee ne ee ee Oe QTE R 
The integration of this equation with reference to v, leads to an expression for e, 
involving an arbitrary function of ¢, for the determination of which more data 
from experiment are required. It would, for instance, be sufficient for this pur- 
pose, to have the mechanical energy of the fluid for all temperatures when con- 
tained in a constant volume; or, what amounts to the same (it being now supposed 
that J is known), to have the thermal capacity of the fluid in constant volume, for 
a particular volume and all temperatures. Hence, we conclude, that when the 
elements J and p belonging to the general theory of the mechanical action of heat 
are known, the mechanical energy of a particular fluid may be investigated with- 
out experiment, from determinations of its pressure for all temperatures and 
volumes, and its thermal capacity for any particular constant volume and all tem- 
peratures. 
90. For example, let the fluid be atmospheric air, or any other subject to the 
“gaseous” laws. Then if % be the volume of a unit of weight of the fiuid, and 0 
the temperature, in the standard state from which the mechanical energy in any 
other state is reckoned, and if p, denote the corresponding pressure, we have 
=o" dp _ py» HE 
poh +E), Po M% 
J dp = JE o2 
and - Ga —p) dv=p, % {<- (+89 } log 
Hence, if we denote by N, the value of N when v=, whatever be the tempera- 
ture, we have, as the general expression for the mechanical energy of a unit weight 
of a fluid subject to the gaseous laws, 
e=p, % {7-a+B9 } tog S+af' Nat AaB ras) 551i: (0); 
91. Let us now suppose the mechanical energy of a particular fluid mass in 
various states to have been determined in any way, and let us find what results 
regarding its pressure and thermal capacities may be deduced. In the first place, 
by integrating equation (8), Gousttered as a differential equation with reference to 
t, for p, we find 
t 1 st 
1 
shiva erg, nine sf, wat 
p=e’ f nite uh dt+ p(w) €° 5 si pS tye tee? 
where ¥ (v) denotes a constant with reference to ¢, which may vary with v, and 
cannot be determined without experiment. Again, we have, from (5), (4), and (1), 
lde 
LTTE 
dp 
— ; i : 5 ; 11). 
caldel(de,,) ae f° Ge 
Fade rae dv _4ap 
dv 
VOL. XX. PART III. 6N 
