Page:Scientific Memoirs, Vol. 1 (1837).djvu/388

We thus see that $\phi (pv)$ is given by a series of terms, each of which is obtained by means of the preceding one, by differentiating it in respect to $v$, multiplying by the ratio ${\frac {\frac {d\,T}{d\,p}}{\frac {d\,T}{d\,v}}}$, and integrating the result in respect of $p$. The first term of this series being $\int {\frac {d\,p}{\frac {d\,T}{d\,v}}}$, it is evident that the value of $\phi$ may be easily obtained; substituting this value in the equation (1), we have for the expression of the general integral of the partial differential equation

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

the formula

$Q=F(T)-C\left|{\begin{aligned}&\int {\frac {d\,p}{\frac {d\,T}{d\,v}}}\\&+\int d\,p{\frac {\frac {d\,T}{d\,p}}{\frac {d\,T}{d\,v}}}{\frac {d}{d\,v}}\int {\frac {d\,p}{\frac {d\,T}{d\,v}}}\\&+\int d\,p{\frac {\frac {d\,T}{d\,p}}{\frac {d\,T}{d\,v}}}{\frac {d}{d\,v}}\int d\,p{\frac {\frac {d\,T}{d\,p}}{\frac {d\,T}{d\,v}}}{\frac {d}{d\,v}}\int {\frac {d\,p}{\frac {d\,T}{d\,v}}}\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \end{aligned}}\right.$

This equation gives the law of the specific calorics, and of the heat disengaged by the variations of the volume and of the pressure of all the substances of nature, when the relation which exists between the temperature, the volume, and the pressure is known.