in the third operation must be also equal to dv, or only differ from it by an infinitely small quantity of the second order. During the second operation we may suppose the volume to be increased by an infinitely small quantity ϕ; which will occasion a diminution of pressure and a diminution of temperature, denoted respectively by ω and τ. During the fourth operation there will be a diminution of volume and an increase of pressure and temperature, which can only differ, by infinitely small quantities of the second order, from the changes in the other direction, which took place in the second operation, and they also may, therefore, be denoted by ϕ, ω, and τ, respectively. The alteration of pressure
function, of two independent variables v and t, is merely an analytical expression of Carnot's fundamental axiom, as applied to a mass of air. The general principle may be analytically stated in the following terms:—If Mdv denote the accession of heat received by a mass of any kind, not possessing a destructible texture, when the volume is increased by dv, the temperature being kept constant, and if Ndt denote the amount of heat which must be supplied to raise the temperature by dt, without any alteration of volume; then Mdv + Ndt must be the differential of a function of v and t. [Note of Nov. 5, 1881. In the corrected theory it is (M - Jp)dv + Ndt, that is a complete differential, not Mdv + Ndt. See Dynamical Theory of Heat (Art. xlviii., below), § 20.]
