Part 37

Footnote 42:

A case minutely examined in another paper, to be laid before the Society at the present meeting. “Theoretical Considerations on the Effect of Pressure in Lowering the Freezing-point of Water,” by Prof. James Thomson.

Footnote 43:

In all that follows, the pressure of the atmosphere on the upper side of the piston will be included in the applied forces, which, in the successive operations described, are sometimes overcome by the upward motion, and sometimes yielded to in the motion downwards. It will be unnecessary, in reckoning at the end of a cycle of operations, to take into account the work thus spent upon the atmosphere, and the restitution which has been made, since these precisely compensate for one another.

Footnote 44:

let the piston be pushed down to any position E{3}F{3};

then Carnot’s fourth operation altered to the following:

let the piston be pushed down from E{3}F{3} until the temperature reaches its primitive value S;

and lastly, Carnot’s first operation altered to the following:

let the piston rise to its primitive position.]

Footnote 45:

In Carnot’s work some perplexity is introduced with reference to the temperature of the water, which, in the operations he describes, is not brought back exactly to what it was at the commencement; but the difficulty which arises is explained by the author. No such difficulty occurs with reference to the cycle of operation described in the text, for which I am indebted to Mons. Clapeyron.

Footnote 46:

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.]

Footnote 47:

We might also investigate another relation, to express the fact that there is no accession or removal of heat during either the second or the fourth operation; but it will be seen that this will not affect the result in the text, although it would enable us to determine both φ and ω in terms of τ.

Footnote 48:

This result might have been obtained by applying the usual notation of the integral calculus to express the area of the curvilinear quadrilateral, which, according to Clapeyron’s graphical construction, would be found to represent the entire mechanical effect gained in the cycle of operations of the air-engine. It is not necessary, however, to enter into the details of this investigation, as the formula (3), and the consequences derived from it, include the whole theory of the air-engine, in the best practical form; and the investigation of it which I have given in the text will probably give as clear a view of the reasoning on which it is founded as could be obtained by the graphical method, which in this case is not so valuable as it is from its simplicity in the case of the steam-engine.

Footnote 49:

This paragraph is the demonstration, referred to above, of the proposition stated in § 13, as it is readily seen that it is applicable to any conceivable kind of thermodynamic engine.

Footnote 50:

The results of these investigations are exhibited in Tables I and II.

Footnote 51:

It is, comparatively speaking, of little consequence to know accurately the value of σ, for the factor (1 − σ) of the expression for μ, since it is so small (being less than ¹⁄₁₇₀₀ for all temperatures between 0° and 100°) that, unless all the data are known with more accuracy than we can count upon at present, we might neglect it altogether, and take dp/kdt simply, as the expression for μ, without committing any error of important magnitude.

Footnote 52:

This is well established, within the ordinary atmospheric limits, in Regnault’s Études Météorologiques, in the Annales de Chimie, vol. xv., 1846.

Footnote 53:

It appears that the vol. of 1 kilog. must be 1.69076 according to the data here assumed.