Page:Reflections on the Motive Power of Heat.djvu/182

for the amounts gained during the first and second, and spent during the third and fourth operations; and hence, by addition and subtraction, we find

$\omega dv-p\phi {\dfrac {dv}{v}}$, or $(v\omega -p\phi ){\dfrac {dv}{v}}$,

for the aggregate amount of mechanical effect gained during the cycle of operations. It only remains for us to express this result in terms of dq and τ, on which the given thermal agency depends. For this purpose we remark that ϕ and ω are alterations of volume and pressure which take place along with a change of temperature τ, and hence, by the laws of compressibility and expansion, we may establish a relation^[1] between them in the following manner:

Let p₀ be the pressure of the mass of air when reduced to the temperature zero, and confined in a volume v₀; then, whatever be v₀, the product p₀v₀ will, by the law of compressibility, remain constant; and, if the temperature be elevated from 0 to t + τ and the gas be allowed to expand freely without any change of pressure, its volume will be

1. We might also investigate another relation, to express the fact that there is no accession or removal of heat during either the second or the fourth operation; but it will be seen that this will not affect the result in the text, although it would enable us to determine both ϕ and ω in terms of τ.