480 PROFESSOR WILLIAM THOMSON ON THE 
From the first of these equations we infer that with a complete knowledge of 
the mechanical energy of a particular fluid, we have enough of data for determining 
for every state, its thermal capacity in constant volume. From equation (9) we 
infer, that with, besides, a knowledge of the pressure for all volumes and a parti- 
cular temperature, or for all volumes and a particular series of temperatures, we 
have enough to determine completely the pressure, and consequently also, accord- 
ing to equation (11), to determine the two thermal capacities, for all states of the 
fluid. 
92. For example, let these equations be applied to the case of a fluid subject 
to the gaseous laws. If we use for se its value derived from (9), in equation (10), 
we find. 
1 wadt 
<= J 
p= AL EDtx@e” Jv a 
where x (v), denoting an arbitrary function of v, is used instead of ¥ (v) — Poo" 
We conclude that the same expression for the mechanical energy holds for any 
fluid whose pressure is expressed by this equation, as for one subject to the gase- 
5 : de de 3 : 5 
ous laws. Again, by using for aie and qe their values derived from (9), in equa- 
tion (11), we have 
d {=- (a+Bd } 
1 v 
N=N,+ FZ Po% ae (18), 
K=N, fe Py % logs — : . (14). 
The first of these equations shews that, unless Mayer’s hypothesis be true, there 
is a difference in the thermal capacities in constant volume, of the same gas at the 
same temperatures for different densities, proportional in amount to the difference 
of the logarithms of the densities. The second compared with the first, leads to 
an expression for the difference between the thermal capacities of a gas in constant 
volume, and under constant pressure, agreeing with results arrived at formerly. 
(Account of Carnot’s Theory, Appendix iii., and Dyn. Th. of Heat, § 48.) 
93. It may be, that more or less information, regarding explicitly the pressure 
and thermal capacities of the fiuid, may have been had as the data for determining 
the mechanical energy ; but these converse deductions are still interesting, as shew- 
ing how much information regarding its physical properties, is comprehended in 
a knowledge of the mechanical energy of a fluid mass, and how useful a table of 
the values of this function for different temperatures and volumes, or an Empirical 
Function of two variables expressing it, would be, whatever be the experimental 
