∫^{_p_{1}_}_{_p_{3}_}(1 - σ)(_H_)/(_k_)_dp_.
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_ ∫_{_T_}^{_S_}(1 - σ)((_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
− σ)(_dp_)/(_dt_/_k_) will not sensibly vary for values of _t_ between
_T_ and _S_,—the preceding expression becomes simply
_Μ_ = (1 - σ)(_dp_)/(_dt_)/(_k_). _Η_(_S_ - _Τ_). (2)
This might, of course, have been obtained at once by supposing the
breadth of the quadrilateral figure _AA_{1}A_{2}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 the line _AA_{1}_, or
the line _A_{3}A_{2}_, or any line of intermediate magnitude.
The expression (2) is rigorously correct for any interval _S_ − _T_, if
the mean value of (1 − σ)((_dp_/_dt_)/_k_) for that interval be employed
as the coefficient of _H_(_S_ − _T_).