^{page 261 note *} From Davy's, first work, entitled “An Essay on Heat, Light, and the Combinations of Light,” published in 1799, in “Contributions to Physical and Medical Knowledge, principally from the West of England, collected by M.D.Beddoes, Thomas,” and republished in Dr Davy's edition of his brother's collected works, vol. ii. Lond. 1836.Google Scholar

^{page 261 note †} In May 1842, Mayer announced in the “Annalen” of Wöhler and Liebig, that he had raised the temperature of water from 12° to 13° cent. by agitating it. In August 1843, Joule announced to the British Association, “That heat is evolved by the passage of water through narrow tubes;” and that he had “obtained one degree of heat per lb. of water from a mechanical force capable of raising 770 lbs. to the height of one foot;” and that heat is generated when work is spent in turning a magneto-electric machine, or an electro-magnetic engine. (See his paper “On the Calorific Effects of Magneto-Electricity, and on the Mechanical Value of Heat.” Phil. Mag. vol. xxiii. 1843Google Scholar.)

^{page 262 note *} “Annalen” of Wöhler and Liebig, May 1842.

^{page 262 note †} British Association, August 1843, and Philosophical Magazine, September 1843.

^{page 265 note *} If this axiom be denied for all temperatures, it would have to be admitted that a self-acting machine might be set to work and produce mechanical effect by cooling the sea or earth, with no limit but the total loss of heat from the earth and sea, or, in reality, from the whole material world.

^{page 266 note *} “Account of Carnot's Theory,” § 13.

^{page 266 note †} Ibid., § 6.

^{page 266 note ‡} Poggendorff's Annalen, referred to above.

^{page 267 note *} “There are [at present known] two, and only two, distinct ways in which mechanical effect can be obtained from heat. One of these is by the alterations of volume which bodies experience through the action of heat, the other is through the medium of electric agency.”—Account of Carnot's Theory, § 4. (Transactions, Vol. XVI., Part V.) ……A paper by Mr Joule, containing demonstrations of these laws, and of others on the relations of the chemical and thermal agencies concerned, was communicated to the Royal Society on the 17th December 1840, but was not published in the Transactions. (See abstract containing a statement of the laws quoted above, in the Philosophical Magazine, vol. xviii., p. 308). It was published in the Philosophical Magazine in October 1841 (vol. xix., p. 260).

^{page 267 note †} That, in a given fixed part of the circuit, the heat evolved in a given time is proportional to the square of the strength of the current, and for different fixed parts, with the same strength of current, the quantities of heat evolved in equal times are as the resistances.

^{page 267 note ‡} This reasoning was suggested to me by the following passage contained in a letter which I received from Mr Joule on the 8th of July 1847. “In Peltier's experiment on cold produced at the bismuth and antimony solder, we have an instance of the conversion of heat into the mechanical force of the current,” which must have been meant as an answer to a remark I had made, that no evidence could be adduced to shew that heat is ever put out of existence. I now fully admit the force of that answer, but it would require a proof that there is more heat put out of existence at the heated soldering than is created at the cold soldering, to make the “evidence” be experimental. That this is the case I think is certain, because the statements of § 16 in the text are demonstrated consequences of the first fundamental proposition; but it is still to be remarked, that neither in this nor in any other case of the production of mechanical effect from purely thermal agency, has the ceasing to exist of an equivalent quantity of heat been demonstrated otherwise than theoretically. It would be a very great step in the experimental illustration (or verification, for those who consider such to be necessary) of the dynamical theory of heat, to actually shew, in any one case, a loss of heat: and it might be done by operating through a very considerable range of temperatures with a good air-engine or steam-engine, not allowed to waste its work in friction. As will be seen in Part II. of this paper, no experiment of any kind could shew a considerable loss of heat without employing bodies differing considerably in temperature; for instance, a loss of as much as ·098, or about one-tenth of the whole heat used, if the temperature of all the bodies used be between 0° and 30° cent.

^{page 269 note *} “Account of Carnot's Theory,” foot-note on § 26.

^{page 269 note †} This may have parts consisting of different substances, or of the same substance in different states, provided the temperature of all be the same. See below Part III., §§ 53–56.

^{page 273 note *} “Account,” &c, Equation 7, § 31.

^{page 273 note †} “Account,” &c. Appendix, Section IV.

^{page 274 note *} “On the Changes of Temperature produced by the Rarefaction and Condensation of Air,” Phil. Mag., vol. xxvi. May 1845Google Scholar.

^{page 275 note *} See below (Part III. § 58), where the “negative” specific heat of saturated steam is investigated. If the mean value of this quantity between 0° and 100° were – 1·5 (and it cannot differ much from this) there would be 150 units of heat emitted by a pound of saturated vapour in having its temperature raised (by compression) from 0° to 100°. The latent heat of the vapour at 0° being 606·5, the final quantity of heat required to convert a pound of water at 0° into saturated steam at 100°, in the first of the ways mentioned in the test, would consequently be 456·5, which is only about of the quantity 637 found as “the total heat” of the saturated vapour at 100°, by Regnault.

^{page 276 note *} If the steam have to rush through a long fine tube, or through a small aperture within the calorimetrical apparatus, its pressure will be diminished before it is condensed, and there will, therefore, in two parts of the calorimeter be saturated steam at different temperatures (as, for instance, would be the case if steam from a high pressure boiler were distilled into the open air); yet, on account of the heat developed by the fluid friction, which would be precisely the equivalent of the mechanical effect of the expansion wasted in the rushing, the heat measured by the calorimeter would be precisely the same as if the condensation took place at a pressure not appreciably lower than that of the entering steam. The circumstances of such a case have been overlooked by Clausius (Poggendorff's Annalen, 1850, No. 4, p. 510), when he expresses with some doubt the opinion that the latent heat of saturated steam will be truly found from Regnault's “total heat,” by deducting the sensible heat; and gives as a reason that, in the actual experiments, the condensation must have taken place “under the same pressure, or nearly under the same pressure,” as the evaporation. The question is not, Did the condensation take place at a lower pressure than that of the entering steam? but, Did Regnault make the steam work an engine in passing through the calorimeter, or was there so much noise of steam rushing through it as to convert an appreciable portion of the total heat into external mechanical effect? And a negative answer to this is a sufficient reason for adopting with certainty the opinion that the principle of his determination of the latent heat is correct.

^{page 277 note *} I cannot see that any hypothesis, such as that adopted by Clausius fundamentally in his investigations on this subject, and leading, as he shews, to determinations of the densities of saturated steam at different temperatures, which indicate enormous deviations from the gaseous laws of variation with temperature and pressure, is more probable, or is probably nearer the truth, than that the density of saturated steam does follow these laws as it is usually assumed to do. In the present state of science it would perhaps be wrong to say that either hypothesis is more probable than the other.

^{page 277 note †} It ought to be remarked that, as the unit of force implied in the determinations of μ is the weight of a pound of matter at Paris, and the unit of force in terms of which J is expressed is the weight of a pound at Manchester, these numbers ought in strictness to be modified so as to express the values in terms of a common unit of force; but as the force of gravity at Paris differs by less than of its own value from the force of gravity at Manchester, this correction will be much less than the probable errors from other sources, and may therefore be neglected.

^{page 280 note *} If we take where k may be any constant, we find

;

which is the formula I gave when this paper was communicated. I have since remarked, that Mr Joule's hypothesis implies essentially, that the coefficient k must be as it is taken in the text, the mechanical equivalent of a thermal unit. Mr Rankine, in a letter dated March 27, 1851, informs me that he has deduced, from the principles laid down in his paper communicated last year to this Society, an approximate formula for the ratio of the maximum quantity of heat converted into mechanical effect to the whole quantity expended, in an expansive engine of any substance, which, on comparison, I find agrees exactly with the expression (12) given in the text as a consequence of the hypothesis suggested by Mr Joule regarding the value of μ at any temperature.—[April 4,1851.]

^{page 281 note *} See above, § 22.

^{page 286 note *} Transactions, Vol. xvi., Part v. His paper was republished, with some slight modifications, in the Cambridge and Dublin Mathematical Journal, New Series, Vol. V.—Nov. 1850.

^{page 287 note *} This explanation has been objected to as incorrect in principle by Clausius, in an article recently published in Poggendorff's Annalen. I trust that, on reconsidering the subject (and, should this meet his eye, on reading the statement in the text, and the remarks in § 33 above), he will perceive that my explanation, as originally stated, is perfectly correct.