Page:Reflections on the Motive Power of Heat.djvu/183

increased in the ratio of 1 to 1 + E(t + τ), where E is very nearly equal to .00366 (the Centigrade scale of the air-thermometer being referred to), whatever be the gas employed, according to the researches of Regnault and of Magnus on the expansion of gases by heat. If, now, the volume be altered arbitrarily with the temperature continually at t + τ, the product of the pressure and volume will remain constant; and therefore we have

$pv=p_{_{0}}v_{_{0}}\{1+E(t+\tau )\}.$

Similarly,

$(p-\omega )(v+\phi )=p_{_{0}}v_{_{0}}\{1+Et\}.$

Hence, by subtraction, we have

$v\omega -p\phi +\omega \phi =p_{_{0}}v_{_{0}}E\tau ,$

or, neglecting the product ωϕ,

$v\omega -p\phi =p_{_{0}}v_{_{0}}E\tau .$

Hence the preceding expression for mechanical effect, gained in the cycle of operations, becomes

$p_{_{0}}v_{_{0}}.E\tau .dv/v.$

Or, as we may otherwise express it,

${\frac {Ep_{_{0}}v_{_{0}}}{vdq/dv}}.dq.\tau .$

Hence, if we denote by M the mechanical effect due to H units of heat descending through the same interval τ, which might be obtained by repeating