^{page 542 note *} Carnot, p. 37.

^{page 543 note *} The evolution of heat in a fixed conductor, through which a galvanic current is sent from any source whatever, has long been known to the scientific world; but it was pointed out by Mr Joule that we cannot infer from any previously-published experimental researches, the actual generation of heat when the current originates in electro-magnetic induction; since the question occurs, is the heat which is evolved in one part of the closed conductor merely transferred from those parts which are subject to the inducing influence ? Mr Joule, after a most careful experimental investigation with reference to this question, finds that it must be answered in the negative.—(See a paper “ On the Calorific Effects of Magneto-Electricity, and on the Mechanical Value of Heat; by J. P. Joule, Esq.” Read before the British Association at Cork in 1843, and subsequently communicated by the Author to the Philosophical Magazine, vol. xxiii., pp. 263, 347, 435.)

Before we can finally conclude that heat is absolutely generated in such operations, it would be necessary to prove that the inducing magnet does not become lower in temperature, and thus compensate for the heat evolved in the conductor. I am not aware that any examination with reference to the truth of this conjecture has been instituted; but, in the case where the inducing body is a pure electro-magnet (without any iron), the experiments actually performed by Mr Joule render the conclusion probable that the heat evolved in the wire of the electro-magnet is not affected by the inductive action, otherwise than through the reflected influence which increases the strength of its own current.

^{page 544 note *} So generally is Carnot's principle tacitly admitted as an axiom, that its application in this case has never, so far as I am aware, been questioned by practical engineers.

^{page 545 note *} When “thermal agency” is thus spent in conducting heat through a solid, what becomes of the mechanical effect which it might produce ? Nothing can be lost in the operations of nature—no energy can be destroyed. What effect then is produced in place of the mechanical effect which is lost ? A perfect theory of heat imperatively demands an answer to this question; yet no answer can be given in the present state of science. A few years ago, a similar confession must have been made with reference to the mechanical effect lost in a fluid set in motion in the interior of a rigid closed vessel, and allowed to come to rest by its own internal friction; but in this case, the foundation of a solution of the difficulty has been actually found, in Mr Joule's discovery of the generation of heat, by the internal friction of a fluid in motion. Encouraged by this example, we may hope that the very perplexing question in the theory of heat, by which we are at present arrested, will, before long, be cleared up.

It might appear, that the difficulty would be entirely avoided, by abandoning Carnot's fundamental axiom; a view which is strongly urged by Mr Joule (at the conclusion of his paper “On the Changes of Temperature produced by the Rarefaction and Condensation of Air.” Phil. Mag., May 1845, vol. xxvi.) If we do so, however, we meet with innumerable other difficulties—insuperable without farther experimental investigation, and an entire reconstruction of the theory of heat, from its foundation. It is in reality to experiment that we must look—either for a verification of Carnot's axiom, and an explanation of the difficulty we have been considering; or for an entirely new basis of the Theory of Heat.

^{page 545 note †} For a demonstration, see § 29, below.

^{page 546 note *} A case minutely examined in another paper, to be laid before the Society at the present meeting.

^{page 547 note *} 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.

^{page 547 note †} 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 operations described in the text, for which I am indebted to Mons. Clapeyron.

^{page 549 note *} See Note at the end of this Paper.

^{page 551 note *} Thus, 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 M. dv denote the accession of heat received by a mass of any kind, not possessing a destructible texture, when the volume is increased by d v, the temperature being kept constant, and if N d t denote the amount of heat which must be supplied to raise the temperature by d t, without any alteration of volume; then M dv + N dt must be the differential of a function of v and t.

^{page 553 note *} 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.

^{page 554 note *} 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.

^{page 555 note *} 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 thermo-dynamic engine.

^{page 557 note *} The results of these investigations are exhibited in Tables I. and II. below.

^{page 557 note †} 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 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 simply, as the expression for μ, without committing any error of important magnitude.

^{page 557 note ‡} This is well established, within the ordinary atmospheric limits, in Regnault's Études Météorologiques, in the Annales de Chimie, vol. xv., 1846.

^{page 558 note ‡} The part of this expression in the first vinculum (see Regnault, end of ninth Mémoire) is what is known as “the total heat” of a pound of steam, or the amount of heat necessary to convert a pound of water at 0° into a pound of saturated steam at t°; which, according to “Watt's law,” thus approximately verified, would be constant. The second part, which would consist of the single term t, if the specific heat of water were constant for all temperatures, is the number of thermic units necessary to raise the temperature of a pound of water from 0° to t°, and expresses empirically the results of Regnault's experiments on the specific heat of water (see end of the tenth Mémoire), described in the work already referred to.

^{page 559 note *} In strictness, the 230th is the last degree for which the experimental data are complete; but the data for the 231st may readily be assumed in a sufficiently satisfactory manner.

^{page 560 note *} The numbers here tabulated may also be regarded as, the actual values of μ for t=½, t = 1½, t=2½, t=3½, &c.

^{page 562 note *} For, at the end of the fourth operation, the whole mass is liquid, and at the temperature t. Now, this state might be arrived at by first compressing the vapour into water at the temperature t, and then raising the temperature of the liquid to S; and however this state may be arrived at, there cannot, on the whole, be any heat added to or subtracted from the contents of the cylinder, since, during the fourth operation, there is neither gain nor loss of heat. This reasoning is, of course, founded on Carnot's fundamental principle, which is tacitly assumed in the commonly-received ideas connected with “Watt's law,” the “latent heat of steam,” and “the total heat of steam.”

^{page 565 note *} Thus, from Carnot's calculations, we find, in the case of alcohol, 4·035; and in the case of water, 3·648, instead of 3·963, and 3·658, which are Clapeyron's results in the same cases.

^{page 566 note *} A still closer agreement must be expected, when more accurate experimental data are afforded with reference to the other media. Mons. Regnault informs me that he is engaged in completing some researches, from which we may expect, possibly before the end of the present year, to be furnished with all the data for five or six different liquids which we possess at present for water. It is therefore to be hoped that, before long, a most important test of the validity of Carnot's theory will be afforded.

^{page 566 note †} The Napierian logarithm of is here understood.

^{page 567 note *} Carnot varies the statement of his theorem, and illustrates it in a passage, pp. 52, 53, of which the following is a translation:—

“ When a gas varies in volume without any change of temperature, the quantities of heat absorbed or evolved by this gas are in arithmetical progression, if the augmentation or diminutions of volume are in geometrical progression.

“ When we compress a litre of air maintained at the temperature 10°, and reduce it to half a litre, it disengages a certain quantity of heat. If, again, the volume be reduced from half a litre to a quarter of a litre, from a quarter to an eighth, and so on, the quantities of heat successively evolved will be the same.

“ If, in place of compressing the air, we allow it to expand to two litres, four litres, eight litres, &c, it will be necessary to supply equal quantities of heat to maintain the temperature always at the same degree.”

^{page 569 note *} Or the capacity of a unit of volume for heat.

^{page 570 note *} Carnot suggests a combination of the two principles, with air as the medium for receiving the heat at a very high temperature from the furnace; and a second medium, alternately in the state of saturated vapour and liquid water, to receive the heat, discharged at an intermediate temperature from the air, and transmit it to the coldest part of the apparatus. It is possible that a complex arrangement of this kind might be invented, which would enable us to take the heat at a higher temperature, and discharge it at a lower temperature than would be practicable in any simple air-engine or simple steam-engine. If so, it would no doubt be equally possible, and perhaps more convenient, to employ steam alone, but to use it at a very high temperature not in contact with water in the hottest part of the apparatus, instead of, as in the steam-engine, always in a saturated state.

^{page 570 note †} It is probably this invention to which Carnot alludes in the following passage (p. 112): — “Il a été fait, dit-on, tout recemment en Angleterre des essais heureux sur le développement de la puissance motrice par l'action de la chaleur sur l'air atmosphérique. Nous ignorons entièrement ne quoi ces essais ont consisté, si toutefois ils sont réels.”

^{page 570 note ‡} From this point of view, we see very clearly how imperfect is the steam-engine, even after all Watt's improvements. For to “push the principle of expansion to the utmost,” we must allow the steam, before leaving the cylinder, to expand until its pressure is the same as that of the vapour in the condenser. According to “Watt's law,” its temperature would then be the same as (actually a little above, as Regnault has shewn) that of the condenser, and hence the steam-engine worked in this most advantageous way, has in reality the very fault that Watt found in Newcomen's engine. This defect is partially remedied by Hornblower's system of using a separate expansion cylinder, an arrangement, the advantages of which did not escape Carnot's notice, although they have not been recognised extensively among practical engineers, until within the last few years.

^{page 571 note *} I am indebted to the kindness of Professor Gordon of Glasgow, for the information regarding the various cases given in the text.

^{page 571 note †} In different Cornish engines, the pressure in the boiler is from 2½ to 5 atmospheres; and, therefore, as we find from Regnault's table of the pressure of saturated steam, the temperature of the water in the boiler must, in all of them, lie between 128° and 152°. For the better class of engines, the average temperature of the water in the boiler may be estimated at 140°, the corresponding pressure of steam being 3½ temperatures.

^{page 571 note ‡} This number agrees very closely with the number corresponding to the fall from 100° to 0°, given in Table II. Hence, the fall from 140° to 30° of the scale of the air-thermometer is equivalent, with reference to motive power, to the fall from 100° to 0°.

^{page 572 note *} It being assumed that the temperatures of the boiler and condenser are the same as those of the Cornish engines. If, however, the pressure be lower, two atmospheres, for instance, the numbers would stand thus: The temperature in the boiler would be only 121. Consequently, for each pound of steam evaporated, only 614 units of heat would be required; and, therefore, the work performed for each unit of heat transmitted would be 160·3 foot-pounds, which is more than according to the estimate in the text. On the other hand, the range of temperatures, or the fall utilised, is only from 131 to 30, instead of from 140 to 30°, and, consequently (Table II.), the theoretical duty for each unit of heat is only 371 foot-pounds. Hence, if the engine, to work according to the specification, requires a pressure of only 15 lbs. on the square inch (i. e., a total steam pressure of two atmospheres), its performance is , or 43·2 per cent, of its theoretical duty.

^{page 572 note †} If, in this case again, the pressure required in the boiler to make the engine work according to the contract were only 15 lbs. on the square inch, we should have a different estimate of the economy, for which, see Table B, at the end of this paper.

^{page 572 note ‡} These engines are provided with separate expansive cylinders, which have been recently added to them by Mr M'Naught of Glasgow.

^{page 574 note *} Pressure 15 lbs. on the square inch.