Page:Reflections on the Motive Power of Heat.djvu/180

during the first and third operations may at once be determined by means of Mariotte's law, since in them the temperature remains constant. Thus, if, at the commencement of the cycle, the volume and pressure be v and p, they will have become v + dv and pv/(v + dv) at the end of the first operation. Hence the diminution of pressure during the first operation is p - pv/(v + dv) or pdv/(v + dv) and therefore, if we neglect infinitely small quantities of the second order, we have pdv/v for the diminution of pressure during the first operation; which to the same degree of approximation, will be equal to the increase of pressure during the third. If t + τ and t be taken to denote the superior and inferior limits of temperature, we shall thus have for the volume, the temperature, and the pressure at the commencements of the four successive operations, and at the end of the cycle, the following values respectively:

(1)	v,	t + τ,	p;
(2)	v + dv,	t + τ,	p (1 - ${\tfrac {dv}{v}}$);
(3)	v + dv + ϕ,	t,	p (1 - ${\tfrac {dv}{v}}$) - ω;
(4)	v + ϕ,	t,	p - ω;
(5)	v,	t + τ,	p.