Page:Reflections on the Motive Power of Heat.djvu/175

Rh Now, σ, k, and p, being quantities which depend upon the temperature, may be considered as functions of t; and it will be convenient to modify the integral so as to make t the independent variable. The limits will be from t = T to t = S, and, if we denote by M the value of the integral, we have the expression

for the total amount of mechanical effect gained by the operations described above.

21. If the interval of temperatures be extremely small,—so small that $$(1 - \sigma)\frac{\tfrac{dp}{dt}}{k}$$ will not sensibly vary for values of t between T and S, the preceding expression becomes simply

This might, of course, have been obtained at once by supposing the breadth of the quadrilateral figure AA1 A2 A to be extremely small compared with its length, and then taking for its area, as an approximate value, the product of the breadth into