Page:Scientific Memoirs, Vol. 1 (1837).djvu/370

but the temperature remaining constant during the variation of the volume, we have

$vdp+pdv=0,\;\mathrm {whence} \;dp=-{\frac {p}{v}}dv,$

and consequently

$dQ=\left({\frac {dQ}{dv}}-{\frac {p}{v}}{\frac {dQ}{dp}}\right)dv.$

If we divide the effect produced by this value of $dQ$, we shall have

${\frac {R\;dt}{v{\frac {dQ}{dv}}-p{\frac {dQ}{dp}}}}$

for the expression of the maximum effect which can be developed by the passage of a quantity of heat equal to unity, from a body maintained at the temperature $t$ to a body maintained at the temperature $t-dt$.

We have shown that this quantity of action developed is independent of the agent which has served to transmit the heat; it is therefore the same for all the gases, and is equally independent of the ponderable quantity of the body employed: but there is nothing that proves it to be independent of the temperature; $v{\frac {dQ}{dv}}-p{\frac {dQ}{dp}}$ ought therefore to be equal to an unknown function of $t$, which is the same for all the gases.

Now by the equation $pv=R(267+t)$, $t$ is itself the function of the product $pv$; the partial differential equation is therefore

$v{\frac {dQ}{dv}}-p{\frac {dQ}{dp}}=F(p.v),$

having for its integral

$Q=f(p.v)-F(p.v)\log[(hyp)p].$

No change is effected in the generality of this formula by substituting for these two arbitrary functions of the product $pv$, the functions $B$ and $C$ of the temperature, multiplied by the coefficient $R$; we shall thus have

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

That this value of $Q$ satisfies all the conditions to which it is subject may be easily verified; in fact we have

${\begin{aligned}&{\frac {dQ}{dv}}=R\left({\frac {dB}{dt}}{\frac {p}{R}}-\log p{\frac {dC}{dt}}{\frac {p}{R}}\right)\\&{\frac {dQ}{dp}}=R\left({\frac {dB}{dt}}{\frac {v}{R}}-\log p{\frac {dC}{dt}}{\frac {v}{r}}-C{\frac {1}{p}}\right);\end{aligned}}$