Page:Reflections on the Motive Power of Heat.djvu/181

Taking the mean of the pressures at the beginning and end of each operation, we find

(1)	$p{\big (}1-{\tfrac {1}{2}}{\tfrac {dv}{v}}{\big )},$
(2)	$p{\big (}1-~~{\tfrac {dv}{v}}{\big )}-{\tfrac {1}{2}}\omega ,$
(3)	$p{\big (}1-{\tfrac {1}{2}}{\tfrac {dv}{v}}{\big )}-\omega ,$
(4)	$p-{\tfrac {1}{2}}\omega ,$

which, as we are neglecting infinitely small quantities of the second order, will be the expressions for the mean pressures during the four successive operations. Now, the mechanical effect gained or spent, during any of the operations, will be found by multiplying the mean pressure by the increase or diminution of volume which takes place; and we thus find

(1)	$~~p{\big (}1-{\tfrac {1}{2}}{\tfrac {dv}{v}}{\big )}dv,$
(2)	${\big \{}p{\big (}1-~~{\tfrac {dv}{v}}{\big )}-{\tfrac {1}{2}}\omega {\big \}}\phi ,$
(3)	${\big \{}p{\big (}1-{\tfrac {1}{2}}{\tfrac {dv}{v}}{\big )}-\omega {\big \}}dv,$
(4)	${\big (}p-{\tfrac {1}{2}}\omega {\big )}\phi .$