Page:Scientific Memoirs, Vol. 1 (1837).djvu/374

of the liquid corresponding to the temperature $t$ of the body $A$, and $f\,g$ that which corresponds to the temperature $t-d\,t$ of the body $B$; $b\,h$ the increase of volume due to the vapour formed in contact with the body $A$, $h\,k$ that which is due to the vapour formed after the body $A$ has been removed, the formation of which has reduced the temperature by the quantity $d\,t$; we have seen, I say, that the quantity of action developed by the transmission of the latent caloric furnished by the body $A$, [and transmitted] from that body to the body $B$, is measured by the quadrilateral figure $c\,d\,e\,f$. Now this surface is equal, if we neglect the infinitely small quantities of the second order, to the product of the volume $c\,d$ by the differential of the pressure $d\,h-e\,k$. Naming $p$ the pressure of the vapour of the liquid corresponding to the temperature $t$, $p$ will be a function of $t$, and we shall have $d\,h-e\,k={\frac {d\,p}{d\,t}}d\,t$.

$c\,d$ will be equal to the increase of volume produced in water when it passes from the liquid into the gaseous state, under the pressure $p$, at a corresponding temperature. If we call $\rho$ the density of the liquid, $\delta$ that of the vapour, and $v$ the volume of the vapour formed, $\delta v$ will be its weight, and ${\frac {\delta v}{\rho }}$ will be the volume of the liquid evaporated. The increase of volume owing to the formation of a volume $v$ of vapour will therefore be

$v\left(1-{\frac {\delta }{\rho }}\right).$

The effect produced will therefore be

$\left(1-{\frac {\delta }{\rho }}\right)v{\frac {d\,p}{d\,t}}d\,t.$

The heat, by means of which this quantity of action has been produced, is the latent caloric of the volume $v$ of vapour formed; let $k$ be a function of $t$ representing the latent caloric contained in the unity of volume of the vapour furnished by the liquid subjected to experiment, at a temperature $t$, and under a corresponding pressure, the latent caloric of the volume $v$ will be $kv$, and the ratio of the effect produced to the heat expended will be expressed by

${\frac {\left(1-{\frac {\delta }{\rho }}\right){\frac {d\,p}{d\,t}}d\,t}{k}}.$

We have demonstrated that it is the greatest which can possibly be obtained; that it is independent of the nature of the liquid employed, and the same as that obtained by the employment of the permanent gases: now we have seen that this is expressed by ${\frac {d\,t}{C}}$, $C$ being a function of $t$ independent of the nature of the gases; we shall therefore also have