REPORTS ON THE STATE OF SCIENCE. On the Present State of our Theoretical and Experimental Knowledge of the Laws of Conduction of Heat. By the Rev. PHILIP KELLAND, M.A., F.R.SS. Lond. and Edin., Prof. of Math. in the University of Edinburgh, late Fellow of Queen's College, Cambridge. THE object of the following report is simply to lay before the Association an outline of the present state of our theoretical knowledge of the law of transmission of heat by conduction, and to examine how far conclusions deduced from theory have been tested by experiment. Reports on the general problem of Radiant Heat have already appeared by Professor Powell*; and on the theoretical laws of conduction and radiation, a portion of the subjectmatter of our present question, Mr. Whewell has briefly touched in his report 'On Magnetism, Electricity, Heat, &c.t' We shall, in consequence, confine ourselves strictly to our immediate limits, noticing only such other branches of the general theory as bear directly or necessarily on the question. We shall avoid all mention of theoretical investigations, however important in themselves, which are not capable of being examined rigidly by direct experiment; nor shall we scruple to pass over the names of a host of illustrious experimenters on conduction and radiation, when we find that their experiments are not calculated to serve as the immediate test of theory. This proceeding will materially shorten our labour, and will have the effect of condensing into a narrow compass all the remarks we have to make. To render what has to be said as clear as possible, the subject-matter has been arranged under three heads. Two of these are distinctly marked out by the statement of the object proposed to be effected, and the third is suggested by a consideration of the former two. We shall examine, then, I. What is the present state of our theoretical knowledge of the phænomena of conduction. We are here to seek for the principles on which the reasoning is based, to inquire what are the axioms of radiation and conduction, or of the flow of heat, which, from observation, experiment or analogy, have been assumed to hold true, and to point out the conclusions to which these axioms have led. We have to distinguish between differing theories, and to contrast with each other some of the most simple of the results to which they respectively lead. This portion of our subject must, to a certain extent, be treated historically. We shall inquire, II. into the state of experimental investigation, so far as it has been undertaken with a view to test or to illustrate the conclusions arrived at by theory. We shall examine how the different consequences of certain hypotheses bear the test thus applied to them, by computing from the * Report on Radiant Heat in Reports of British Association, vols. i. and ix. formulæ the values of the temperature corresponding with the conditions existing in the experiment, and contrasting the results with the temperatures actually observed. This critical discussion of the hypotheses will lead us in the third place To point out, III. the utter inadequacy of the few experimental facts with which we are furnished, to serve either as the basis of a true theory or as the indication of a false one. We admit that, of a theory based on assumptions which have been for a century regarded as only approximative to truth, the experiments are sufficient to expose the incompetency, just as experiments on bodies sliding under the retardation of friction will easily detect the inadequacy of formulæ deduced from the hypothesis of absolute smoothness. But we shall see that, as applicable to point out errors in the assumed axioms on which reasonings are founded to constitute a physical theory, the experiments we possess are defective both in their number and in their nature. We shall find three distinct theories equally verified or equally overturned by them, according as we choose to regard the conclusions as indicating the one or the other; and yet we are quite sure that only one of the theories is the correct one, whilst on the other hand we can hardly entertain a doubt that one of them is so. When we shall have made this appear, it will only remain for us to point out, in conclusion, what are the most important results of theory which it is desirable that experiment should be brought in to test, and to suggest a few of the most simple means of effecting the object desired. I. The problem, in the solution of which consists the mathematical theory of heat, is the following. Having given the state of heating, or the variation of that state from time to time, at one or more points of a homogeneous body of given form and dimensions, to find the permanent or variable temperature at every other point. Thus a ring is kept at a certain temperature at one point, and it is proposed to discover, 1. what is the variation from time to time of the temperature at every other point, and 2. what is the ultimate temperature to which that at any given point approaches as the time during which the constant heating of one point has been kept up is increased. From this statement it will appear that the experimental facts on which the theory must rest are the answers to the following questions. a. According to what law does a heated body lose its temperature to the air, or other medium or space, by which it is surrounded? b. According to what law is temperature transmitted from point to point of a body? On the correctness of the answers which may be assumed as given to these questions depends the applicability of the results obtained to the state of things in nature. But as in mechanics we may reason correctly on assumed laws which are not laws of nature, and obtain conclusions of great importance as approximations to facts, so in the theory of heat the results, although strictly commensurate only with the laws on which they depend, are still highly important even in reference to the things actually existing, differing as they do in certain cases from the expression of the laws. We proceed then to show what answers have been given to the above questions by different theorists, and to explain the evidence on which their truth is supposed to be established. a. Radiation. Sir Isaac Newton appears to have been the first who was led to apply a law of radiation to experiment. The statement of the law is given by him for the first time in a paper in the Philosophical Transactions for 1701*, and is reprinted in his Opusculat. * Philosophical Transactions, 1701, vol. xxii. p. 827. Newton's law of cooling.-The author is constructing a scale of temperatures; he is comparing, for instance, the heat of boiling water with that of the human body. The comparison is made immediately, to the extent to which the thermometer affords an indication of the temperature; beyond this it is requisite to have recourse to some process which involves computation; and to this end Newton admits the hypothesis, to which we apply the designation given above. His words are as follows (translated): "This table was constructed by the use of a thermometer and red-hot iron. By means of a thermometer I found the measure of the heat up to the point at which tin (stannum) is melted, and by heated iron I found the measure of the rest. For the temperature which heated iron communicates to cold bodies contiguous to it, in a given time, is as the total temperature of the iron. Therefore, if the times of cooling are taken in arithmetical progression, the temperatures will be in geometrical progression, and may be found by a table of logarithms." It is affirmed by most modern writers that Newton was led to this law by experiment. This was very probably the case, for to the extent of temperature indicated by his thermometer it would be very nearly verified. The inaccuracy of this law was first pointed out by Martine*. He found, that although it appears very exact when the temperature of the heated body does not differ much from that of the surrounding air, yet when the temperatures differ considerably it is very far from being the case. Erxleben† also proved that the law is at fault in proportion to the excess of the temperature of the body. Mr. Dalton †, in his New System of Chemical Philosophy,' in a truly philosophical manner attempted to re-establish the law of Newton by altering the thermometric scale. The hypothesis on which he bases his views is, that the dilatation of all liquids is subject to the same law. MM. Dulong and Petit conceive that Dalton's views are untenable, arguing that, "even supposing the accuracy of the principles of this new scale to have been proved, we should be constrained to acknowledge that it does not satisfy the condition of rendering the losses of heat of a body proportional to the excess of its temperature above that of the air which surrounds it, or, in other words, that it does not re-establish the law of Richmann §; for it would be necessary in that case that the law of cooling should be the same for all bodies, and our experiments rigorously prove the contrary ||." We presume MM. Dulong and Petit's argument to be based, not, as would appear from the phrase quoted, on the variability of the law of cooling, so much as on the fact that for different substances the two portions whose sum, according to these authors, constitutes the law, are affected with very different multipliers, so that their relative values depend altogether on the nature of the body. To this matter we shall return in the sequel ¶. M. de la Roche of Geneva** likewise pointed out the deviation from Newton's law, at the same time admitting that it is sufficiently accurate to 212° Fahr., which is perhaps rather more than subsequent discoveries warrant us in assuming. We come now to the time when the law was established in its correct form, so far as we can see at present. The whole merit of the discovery is due to * Martine, Essays on Heat, 1740, p. 236, art. 4. † Novi Commentarii, Soc. Gott., vol. viii. p. 74. New System of Chemical Philosophy, 1808, p. 12. § Kraft and Richmann, Novi Commentarii, Petrop. i. p. 195. Dulong and Petit, Journal de l'Ecole Polytechnique, 1820, tom. xi. p. 237. ¶ Consult their Memoir, p. 190. ** Journal de Physique, 1812, tom. lxxv. p. 201, Prop. 6. Annals of Philosophy, vol. ii. P. 100. MM. Dulong and Petit, to whom the Academy of Sciences awarded the prize in 1818, and whose admirable memoir On the Measure of Temperature and the Laws of Communication of Heat' the reader will do well to consult*. All that we can do is to give a very brief outline of their researches. The first step requisite for them to take was the determination of a correct measure of temperature. To present to the eye an indication of the state of heat of a body the principle of dilatation has been most commonly applied, but it becomes a question to ascertain what substance will by its dilatation express the state of heat the most simply. MM. Dulong and Petit, having determined "that all the gases dilate absolutely in the same manner and by the same quantity for the same change of temperature," conclude that the airthermometer is the best indicator of the state of heat. They argue," that the well-known uniformity in the principal physical properties of all the gases, and particularly the perfect identity of the laws of their dilatation, renders it very probable that in this class of bodies the disturbing causes have not the same influence as in solids and liquids, and that, consequently, the changes in volume produced by the action of heat upon them are more immediately dependent on the force which produces them. It is therefore probable (they think) that the greater number of the phænomena relating to heat will present themselves under a more simple form if we measure the temperatures on the air-thermometer. It is at least by these considerations (they inform us) that we have been determined constantly to employ this scalet." Having thus settled that the air-thermometer is to be taken as the measure of temperature, they proceeded in the next place to obtain the laws of cooling in vacuo. And here we cannot but express our regret that the original unreduced observations of the authors are not presented to the world in some work generally known. We have never seen them, nor are we sure that they have been published at all. We take the present opportunity of further expressing our astonishment that experiments on which so much depends have never been repeated in this country. We do not know any more desirable exercise of the funds and energies of public scientific bodies than the repetition of all experiments, and the institution of others in a trying form, on which laws of nature have been partially or totally founded. In the case before us we do not doubt the accuracy or fidelity of the ingenious experiments, but we wish to be assured by cumulative evidence that the constant introduced into their law is determined with sufficient accuracy. To return from this digression. J The velocity of cooling was experimented on by our authors by means of heated thermometers placed in a balloon nearly free from air; but the observations were subjected to two corrections. In the first place the stem of the thermometer without the balloon soon becomes cooled down to the temperature of the surrounding air. Every temperature observed therefore was too low by a number of degrees equal to that to which the mercury in the stem would dilate, when heated from the temperature of the surrounding atmosphere to that of the bulb. A correction on this account was applied to all the temperatures observed. The second correction was destined to reduce the observations actually made on the mercurial thermometer to the corresponding indications of the air-thermometer. Besides these corrections, rendered requisite by the nature of the experiments, there was a third which arose out of the necessary imperfection of the vacuum. This was applied to the resulting velocities, and its value was ascertained by making correspond*Annales de Chimie, tom. vii. p. 225, &c. Thomson's Annals of Philosophy, vol. xiii. p. 113, &c. Journal de l'Ecole Polytechnique, 1820, tom. xi. p. 189. ↑ Journal de l'Ecole Polytechnique, tom. xi. p. 232. ing experiments on vacuums of different degrees of imperfection, and thence computing the amount of error introduced by the action of a known quantity of air. The result to which our authors arrived is expressed by the following law. "When a body cools in vacuo, surrounded by a medium whose temperature is constant, the velocity of cooling for excesses of temperature in arithmetical progression increases as the terms of a geometrical progression, diminished by a constant quantity." The formula which expresses the velocity of cooling is m a' (a3 — 1), where a is the same for all bodies, viz. 1·0077 or 20/1-165, 0 denotes the temperature (marked by the air-thermometer and measured on the centigrade scale) of the vacuum in which the cooling body is placed, and the excess of the temperature of the body above 0. On cooling in air or in gases.-The hypothesis on which was computed the velocity of cooling in air or any other gas, was, that the velocity might be divided into two parts;--the one, that due to direct radiation in vacuo; the other, that due to the actual presence of the gas. The gas was supposed not to influence directly the process of radiation, but to act in aid of it by conduction or convection, or a combination of both. Proceeding thus, MM. Dulong and Petit first verified the observation of Leslie, "that the loss of heat owing to the contact of a gas is independent of the state of the surface of the body which cools." They showed next, "that the velocity of cooling of a body, owing to the sole contact of a gas, depends for the same excess of temperature on the density and temperature of the gas; but this dependence is such that the velocity of cooling remains the same so long as the elasticity is unaltered." They found also, "that the cooling power of a gas is, cæteris paribus, proportional to a certain power of its elasticity, but that the index of the power varies for different gases;" and moreover, "that the velocities of cooling due to a gas increase in geometrical progression as the excesses of temperature increase in geometrical progression." We shall best understand the whole law of cooling by exhibiting it in the shape of a single formula. It is as follows: V = m. 1·0077° (1·0077 — 1) + n e2 1.233, where m depends on the nature of the surface, and n and p on the nature of the gas. O is, as before, the temperature of the gas, and @ + d that of the cooling body; e, the elasticity of the gas. If this be the law of nature, we can hardly term by the same word radiation the loss of heat in vacuo, and the loss due to the action of the surrounding air. We must therefore, for the present, confine our signification of this term to the former, and admit that results deduced from the hypothesis of radiation apply only to experiments carried on in a space free from air. b. Conduction. Ordinary experience teaches us that the power of conduction differs in different substances; and it is natural to suppose, and has, in fact, been universally admitted, that this difference is a difference in intensity only. It is assumed that one and the same law holds good for all bodies, but that a certain factor, on which the absolute amount of conduction depends, differs according to the nature of the substance. But to define the law of conduction, which is the same for all substances, considerable difficulty has been experienced. Lambert*, and the other early writers on the subject, regarded the flow of heat as the flow of a fluid. But when we treat the subject mathematically, and regard the flow of heat as the flow of an * Act. Helvet., vol. ii. p. 172. |