Page:Scientific Memoirs, Vol. 1 (1837).djvu/379

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

such is the quantity of heat consumed in the production of the effect that we have just calculated. The effect produced by a quantity of heat equal to unity will therefore be

${\frac {d\,T}{{\frac {d\,Q}{d\,v}}.\quad {\frac {d\,T}{d\,p}}-{\frac {d\,Q}{d\,p}}.\quad {\frac {d\,T}{d\,v}}.}}$

It will be shown, as in the case of the gases, that this effect produced, is the largest which it is possible to realize; and as all the substances of nature may be employed, in the manner that has just been indicated, to produce this maximum effect, it is necessarily the same for all.

When this theory has been applied specially to the gases, we have called ${\frac {1}{C}}$ the coefficient of $d\,T$ in the expression of this maximum quantity of action; the equation therefore of all the substances of nature, solid, liquid, or gaseous, will be

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

in which $C$ is a function of the temperature which is the same for all.

For the gases we have

$T=-267+{\frac {1}{R}}pv,$

whence we deduce

${\frac {d\,T}{d\,p}}={\frac {v}{R}}\quad {\mbox{and}}\;\;{\frac {d\,T}{d\,v}}={\frac {p}{R}}.$

The preceding equation applied to the gases takes therefore the form

$v{\frac {d\,Q}{d\,v}}-p{\frac {d\,Q}{d\,p}}=RC=F(p,v):$

it is the equation at which we have already arrived, and of which the integral is

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

that of the general equation

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

is of the form

$Q=F(T)-C\phi (p,v);$

$F(T)$ is an arbitrary function of the temperature, and $\phi (p,v)$ a particular function satisfying the equation