Page:Scientific Memoirs, Vol. 1 (1837).djvu/380

${\frac {d\,T}{d\,v}}{\frac {d\,\phi }{d\,p}}-{\frac {d\,T}{d\,p}}{\frac {d\,\phi }{d\,v}}=1$

(See the note appended to this Memoir.)

We shall now deduce various consequences from the general equation at which we have arrived.

We have previously seen that when we compress a body by the quantity $d\,v$, the temperature remaining constant, the heat disengaged by the condensation is equal to

$d\,Q=d\,v\left[{\frac {d\,Q}{d\,v}}-{\frac {d\,Q}{d\,p}}{\frac {\left({\frac {d\,T}{d\,v}}\right)}{\left({\frac {d\,T}{d\,p}}\right)}}\right];$

and as

${\frac {d\,Q}{d\,v}}{\frac {d\,T}{d\,p}}-{\frac {d\,Q}{d\,p}}{\frac {d\,T}{d\,v}}=C,$

the preceding expression takes the form

$d\,Q=d\,v{\frac {C}{\left({\frac {d\,T}{d\,p}}\right)}}=-d\,p{\frac {C}{\left({\frac {d\,T}{d\,v}}\right)}}$

This last equation may be put under the form

$d\,Q=-d\,p\;C{\frac {d\,v}{d\,T}};$

${\frac {d\,v}{d\,T}}$ is the differential coefficient of the volume with regard to the temperature, the pressure remaining constant.

We thus arrive at this general law, which is applicable to all the substances of nature, solid, liquid, or gaseous: If the pressure supported by different bodies, taken at the same temperature, be augmented by a small quantity, quantities of heat will be disengaged from it, which will be proportional to their dilatability by heat.

This result is the most general consequence deducible from this axiom: that it is absurd to suppose that motive force or heat can be created gratuitously and absolutely.

§ VI.

The function of the temperature $C$ is, as we see, of great importance in consequence of the part it sustains in the theory of heat: it enters into the expression of the latent caloric which is contained in all substances, and which is disengaged from them by pressure. Unfortunately no experiments have been made from which we can determine the values of this function, corresponding to all the values of the temperature. To obtain $t=0$ we must proceed in the following manner.