Page:Reflections on the Motive Power of Heat.djvu/273

${\frac {ds}{dt}}$ is simply the specific heat of the gas under constant volume, and our equation (1) is the analytical expression of the law stated on page 86.

If we suppose the specific heat constant at all temperatures (hypothesis discussed above, page 92), the quantity ${\frac {ds}{dt}}$ will be independent of t; and in order to satisfy equation (5) for two particular values of v, it will be necessary that T' and U' be independent of t; we shall then have $T'=C$, a constant quantity. Multiplying T' and C by dt, and taking the integral of both, we find

$T=Ct+C_{1};$

but as $T={\frac {N}{F't}}$, we have

$F't={\frac {N}{T}}={\frac {N}{Ct+C_{1}}}.$

Multiplying both by dt and integrating, we have

$Ft={\frac {N}{C}}\log(Ct+C_{1})+C_{2};$

or changing arbitrary constants, and remarking further that Ft is 0 when t = 0°,

$Ft=A\log {\bigg (}1+{\frac {t}{B}}{\bigg )}.$

(6)

The nature of the function Ft would be thus