Page:Reflections on the Motive Power of Heat.djvu/184

the cycle of operations described above, ${\tfrac {H}{dq}}$ times, we have

$M={\frac {Ep_{_{0}}v_{_{0}}}{vdq/dv}}.H\tau .$

(3)

27. If the amplitudes of the operations had been finite, so as to give rise to an absorption of H units of heat during the first operation, and a lowering of temperature from S to T during the second, the amount of work obtained would have been found to be expressed by means of a double definite integral thus:^[1]

$\left.{\begin{array}{ll}\qquad M=\int _{0}^{H}dq\int _{T}^{S}dt\cdot {\frac {Ep_{_{0}}v_{_{0}}}{vdq/dv}},\\or\\\qquad M=Ep_{_{0}}v_{_{0}}\int _{0}^{H}\int _{T}^{S}{\frac {1}{v}}{\frac {dv}{dq}}\,\cdot {dtdq};\end{array}}\right\}$

(4)

this second form being sometimes more convenient.

1. This result might have been obtained by applying the usual notation of the integral calculus to express the area of the curvilinear quadrilateral, which, according to Clapeyron's graphical construction, would be found to represent the entire mechanical effect gained in the cycle of operations of the air-engine. It is not necessary, however, to enter into the details of this investigation, as the formula (3), and the consequences derived from it, include the whole theory of the air-engine, in the best practical form; and the investigation of it which I have given in the text will probably give as clear a view of the reasoning on which it is founded as could be obtained by the graphical method, which in this case is not so valuable as it is from its simplicity in the case of the steam-engine.