Note 46 — IV.[34] CARNOT’S THEORY OF THE MOTIVE POWER OF HEAT.[35] WITH NUMERICAL RESULTS DEDUCED FROM REGNAULT’S EXPERIMENTS ON STEAM.[36]

Thus, dq/dv will be the partial differential coefficient, with respect to v, of that function of v and t which expresses the quantity of heat that must be added to a mass of air when in a “standard” state (such as at the temperature zero, and under the atmospheric pressure), to bring it to the temperature t and the volume v. That there is such a 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.]