Page:Reflections on the Motive Power of Heat.djvu/174

150 the saturated steam and the water in the cylinder have the same pressure p, and consequently the same temperature, which we may denote by t. Again, throughout the second operation the entire contents of the cylinder possess a greater amount of heat by H units than during the fourth; and, therefore, at any instant of the second operation there is as much more steam as contains H units of latent heat than at the corresponding instant of the fourth operation. Hence if k denote the latent heat in a unit of saturated steam at the temperature t, the volume of the steam at the two corresponding instants must differ by $$\tfrac{H}{k}$$. Now, if σ denote the ratio of the density of the steam to that of the water, the volume $$\tfrac{H}{k}$$ of steam will be formed from the volume $$\sigma\tfrac{H}{k}$$ of water; and consequently we have, for the difference of volumes of the entire contents at the corresponding instants, $$\xi = (1 - \sigma)\frac{H}{k}.$$

Hence the expression for the area of the quadrilateral figure becomes $$\int_{p_3}^{p_1} (1 - \sigma)\frac{H}{k}dp.$$