Page:Reflections on the Motive Power of Heat.djvu/270

If, to abridge, we call N the quantity ${\frac {P}{267}}$, the equation would become

$p=N{\frac {t+267}{v}};$

whence we deduce, according to equation (1),

$dr=N{\frac {t+267}{v}}dv.$

Regarding t as constant, and taking the integral of the two numbers, we shall have

$r=N(t+267)\log v+C.$

If we suppose r = 0 when v = 1, we shall have C = 0; whence

$r=N(t+267)\log v.$

(2)

This is the motive power produced by the expansion of the air which, under the temperature t, has passed from the volume 1 to the volume v. If instead of working at the temperature t we work in precisely the same manner at the temperature t + dt, the power developed will be

$r+\delta r=N(t+dt+267)\log v.$

Subtracting equation (2), we have

$\delta r=N\log vdt.$

(3)

Let e be the quantity of heat employed to maintain the temperature of the gas constant during its