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

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

$M=H\int _{T}^{S}(1-\sigma ){\frac {\frac {dp}{dt}}{k}}\,dt.$

(1)

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

$M=(1-\sigma ){\frac {\frac {dp}{dt}}{k}}.H(S-T).$

(2)

This might, of course, have been obtained at once by supposing the breadth of the quadrilateral figure AA₁A₂A to be extremely small compared with its length, and then taking for its area, as an approximate value, the product of the breadth into