Page:Reflections on the Motive Power of Heat.djvu/173

Rh method of determining the amount of mechanical effect to be obtained from any given thermal agency. As, however, it is merely the area of the figure which it is required to determine, it will not be necessary to be able to describe each of the curves A1 PA2 , A3 P'A, but it will be sufficient to know the difference of the abscissas corresponding to any equal ordinates in the two; and the following analytical method of completing the problem is the most convenient for leading to the actual numerical results.

20. Draw any line PP' parallel to OX, meeting the curvilinear sides of the quadrilateral in P and P'. Let ξ denote the length of this line, and p its distance from OX. The area of the figure, according to the integral calculus, will be denoted by the expression

$$\int_{p3}^{p1}\xi\,dp,$$

where p1  and p3  (the limits of integration indicated according to Fourier's notation) denote the lines OA and N3 A3 , which represent respectively the pressures during the first and third operations. Now, by referring to the construction described above, we see that ξ is the difference of the volumes below the piston at corresponding instants of the second and fourth operations, or instants at which