Page:Scientific Memoirs, Vol. 1 (1837).djvu/371

whence

$v{\frac {dQ}{dv}}-p{\frac {dQ}{dp}}=CR,$

and consequently

${\frac {Rdt}{v{\frac {dQ}{dv}}-p{\frac {dQ}{dp}}}}={\frac {dt}{C}}.$

The function $C$ by which the logarithm of the pressure in the value of $Q$ is multiplied is, as we see, of great importance; it is independent of the nature of the gases, and is a function of the temperature alone; it is essentially positive, and serves as a measure of the maximum quantity of action developed by the heat.

We have seen that of the four quantities $Q$, $t$, $p$, and $v$, two being known, the other two follow from them; they ought therefore to be united together by two equations; one of them,

$pv=R(267+t),$

results from the combined laws of Mariotte and Gay-Lussac. The equation

$Q=R(B-C\log p),$

deduced from our theory, is the second. However, the numerical determination of the alterations produced in the gases, when the volume and the pressure are varied in an arbitrary manner, requires a knowledge of the functions $B$ and $C$.

We shall see upon another occasion that a value approaching to the function $C$ may be obtained through a considerable extent of the thermometrical scale; besides, being determined for one gas it will be determined for all. As to the function $B$, it may vary in different gases; however, it is probable that it is the same for all the simple gases: that they all have the same capacity for heat, is at least the apparent result of the indications of experiment.

Let us return to the equation

$Q=R(B-C\log p).$

We will compress a gas occupying the volume $v$, under the pressure $p$, until the volume becomes $v'$, and allow it to cool till the temperature sinks to the same point. Let $p'$ be the new value of the pressure; let $Q'$ be the new value of $Q$; we shall have

$Q-Q'=RC\log {\frac {p'}{p}}=RC\log {\frac {v}{v'}}.$

The function $C$ being the same for all the gases, it is evident that equal volumes of all the elastic fluids, taken at the same temperature and under the same pressure, being compressed or expanded by the same fraction of their volume, disengage or absorb the same absolute quantity of heat. This law M. Dulong has deduced from direct experiment.