of May last year, in the second part of his paper on the Motive Power of Heat."

§ 4. Holtzmann's objections.

From other quarters repeated and in some cases violent objections were raised to the author's first paper, to which, in the same and following years, a series of other papers, serving to complete the theory, were added. The earliest of these objections came from Holtzmann, who had published in 1845 a short pamphlet* on the subject. In this it would at first appear as if he wished to treat the question from the point of view, that for the generation of work there was necessary not merely a change in the distribution of heat, but also an actual destruction of it, and that conversely by destroying work heat could be again generated. He remarks (p. 7): "The action of the heat which has passed to the gas is thus either a raising of temperature, combined with an increase of the elastic force, or a certain quantity of mechanical work, or a mixture of the two; and a certain quantity of mechanical work is equivalent to the rise in temperature. Heat can only be measured by its effects; of the two above-named effects mechanical work is especially adapted for measurement, and it will be chosen accordingly for the purpose. I call a unit of heat that amount of heat, which by its entrance into a gas can perform the mechanical work a, or, using definite measures, which can raise a kilograms to a height of 1 metre." Further on (p. 12) he determines the numerical value of the constant a by the method previously used by Mayer, and explained in Chapter II., § 5; the number thus obtained corresponds perfectly with the mechanical equivalent of heat, as determined by Joule by various other methods. In extending his theory however, i. e. in developing the equations by which his conclusions are arrived at, he follows the same method as Clapeyron; so that he tacitly retains the assumption that the total quantity of heat which exists is invariable; and therefore that the quantity of heat which a body takes in, while it passes from a given initial condition to its present condition, must be expressible as a function of the variables, which determine that condition.

* Ueber die Wärme und Elasticität der Gase und Dämpfe; von C. Holtzmann. Mannheim, 1845; also Pogg. Ann., Vol. LXXII &.

In the author's first paper the inconsequence of this method was pointed out, and the question treated in another way; on which Holtzmann wrote an article*, in which he endeavoured to shew that this method of treatment, and specially the assumption that heat was expended in producing work, was inadmissible. The first objection which he raised was of a mathematical character. He carried out an investigation similar to that in the author's paper, in order first to determine the excess of the heat which a body takes in over that which it gives out, during a simple cyclical process consisting of indefinitely small variations, and secondly to compare this excess with the work done. But in such a process both the work done and the excess of heat must be indefinitely small quantities of the second order; and therefore in the whole investigation, care must be taken that all quantities of the second order, which do not cancel each other, shall be taken into account. This Holtzmann neglected to do; and he was thus led to a final equation, which contained a self-contradiction, and in which he therefore imagined that he had found a proof of the inadmissibility of the whole method. This objection was easily disposed of by the author in his reply.

He further brought forward as an obstacle to the theory, that, according to the formulæ given, the specific heat of a perfect gas must be independent of its pressure, whereas the experiments of Suermann, and also those of De la Roche and Bérard, shewed that the specific heat of gases increased as the pressure diminished. On this conflict between his own theory and the experiments which were then known and supposed to be correct, the author remarked in his reply as follows: "On this point I must first point out that, even if these observations are perfectly correct, they yet say nothing against the fundamental principle of the equivalence of heat and work, but only against the approximate assumption which I have made, viz. that a permanent gas, if it expand at constant temperature, absorbs only so much heat, as is required for the external work which it thus performs. But besides it is sufficiently known how unreliable are in genera! the determinations of the specific heats of gases; and all the

* Pogg. Ann., Vol. LXXXII. p. 445.

more in those few observations which have been hitherto

made at varying pressures. I did not therefore conceive myself bound to abandon the above assumption on account of these observations, although they were well known to me at the time when I wrote my former work; because the other grounds, which may be alleged for the correctness of the assumption within the limits which I had there laid down, are not wholly destroyed by the grounds which may be alleged against it.”

This remark found its full confirmation in Regnault's Researches, published some years afterwards, on the specific heats of gases, which actually led to the result that these earlier observations were inaccurate, and that the specific heat of permanent gases is not visibly dependent on the pressure.

§ 5. Decher's Objections.

Another most energetic attack on the author's theory was made in 1858, by Professor G. Decher, in a paper "On the Nature of Heat," published in Dingler's Polytechnischer Journal, Vol. 148, pp. 1, 81, 161, 241. He characterizes the author's mathematics, in the first half of his paper of 1850, and in another paper of 1854, as an abuse of analysis, and bungling nonsense; he quotes the equations and principles there cited with single or double notes of admiration, and finally, after proving completely to his own satisfaction that the results are untenable, concludes thus: "These then are the data on which the fundamental principles of the new theory of Heat should rest, and by which its agreement with experience should be proved; they shew in the clearest light that the celebrated work of Herr Clausius, on which he himself and other physicists have built as on a secure foundation, is nothing more than a rotten nut, which looks well from the outside, but in reality contains nothing whatever.”

Of the second half of the paper of 1850, which relates to the second main principle of the theory, Herr Decher observes (p. 163), that having mastered the first half, he saw no inducement to consider the second any further.

On examining more closely the objections raised by Herr Decher against the author's mathematical investigation, it is seen that they are due to the fact that he has not understood

the differential equations there formed, which, though not generally integrable, become so as soon as one further relation is assumed to exist among the variables. In spite of all which the author has said, he has throughout treated the quantities to which these equations refer, viz. the quantities of heat taken in by a body in passing from a given initial condition to its present condition, as mere functions of the variables which determine the condition of the body. After quoting the author's equation for gases, viz.

dQ

dv

dt

d (de) - d (de)

dt

R

= A

..(1),

where A is the heat-equivalent of the unit of work, i.e. the reciprocal of E, he remarks, page 243: "In equation (1) and are fully determined as the differential coefficients of a known function of v and t, viz. Q, taken according to v and t respectively as sole variables; and in whatever way this function may be formed, and whatever relation may be supposed to exist between v and t, the right side of the equation must always equal zero."

This incorrect conception, thus formed by a professed mathematician, convinced the author that the meaning and treatment of this kind of differential equation, although long before established by Monge, was not so generally known as he had supposed; accordingly in his reply*, after a brief notice of some other points raised by Decher, he treated the subject more fully, giving a mathematical explanation, which seemed to him sufficient to obviate any such misunderstandings in future. This was afterwards prefixed to the collection of the author's papers as a mathematical introduction; and the essential part of it has been imported into the mathematical introduction to the present work.

§ 6. Fundamental Principle on which the Author's Proof of the Second Main Principle rests.

The more recent objections to the author's theory, and the departure from his views in more recent treatises, chiefly

refer to his method of proving the second main principle of the theory. This proof rests, as shewn in Chapter III., on the following fundamental principle:-Heat cannot of itself (or without compensation) pass from a colder to a hotter body.

This fundamental principle has been very variously received by the scientific public. Some appear to consider it so self-evident that it is needless to state it as a specific principle, whilst others on the contrary doubt its correctness.

§ 7. Zeuner's first Treatment of the Subject.

The first of the two modes of viewing the question mentioned in the last section appears in Zeuner's valuable paper of 1860 "On the Foundations of the Mechanical Theory of Heat." Here Zeuner gives the author's proof of the second main principle essentially in the same form as it has also been given by Reech. The two differ only in one point. Reech gives the principle, that heat cannot of itself pass from a colder to a hotter body, expressly as a fundamental principle laid down by the author, and bases his proof upon it. Zeuner on the contrary does not mention this principle at all: he shews that if for any two bodies the second main principle of the theory did not hold, then by means of two cyclical processes performed with these two bodies in opposite directions, heat could be made to pass from a colder to a hotter body without any other special change, and he then goes on "as we may repeat both processes as often as we please, using the two bodies alternately in the way described, it would follow that we might, with the aid of nothing and without using either work or heat, continually transfer heat from a body of lower to one of higher temperature, which is an absurdity."

Few readers would probably assent to the opinion that the impossibility of transferring heat from a colder to a hotter body is so self-evident, as is here indicated by the short remark "which is an absurdity." Taking the facts of conduction, and of radiation under ordinary circumstances, we may undoubtedly say that this impossibility is established by daily experience. But even with radiation the question

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Récapitulation très-succincte des recherches algébriques faites sur la théorie des effects mécaniques de la chaleur par différents auteurs: Journ. de Liouville, Ser. II. Vol. 1. p. 58.