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

158 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 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 p0 v0  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