ory of electricity, treated as M. Poisson had treated it in 1811. Mr. Bonnycastle found that all Mr. Barlow's results agreed with such a theory. But a peculiar circumstance in the experiments attracted Mr. Barlow's notice, and made him imagine that the theory required modification. He found that the attraction of a solid iron sphere was the same as that of a hollow shell, even when the shell was thin; and he was led to believe that the magnetic power of iron resides wholly in the surface. This result was confirmed, as to the facts, by Capt. Kater*. Mr. Barlow considered this result as inconsistent with the theory of Coulomb, in which the magnetic fluids in every particle of the mass were supposed to be dislodged by the action of a neighbouring magnet. Yet a little attention shows us that this is in fact a consequence of the Coulombian theory. I have already (p. 11) quoted the passage in Coulomb's memoir on magnetism in which he asserts that the distribution of the sensible magnetism will be the same as if the fluids were transferrible from one part of the body to the other. Now it is easily shown that on this supposition all the sensible magnetism is repelled to the surface, as all the sensible electricity is, according to the parallel theory of electricity, and as Coulomb had shown that it is in fact. It is true, that though the superficial disposition of magnetism followed from the theory, and was involved in the general proposition above quoted, I do not know that Coulomb anywhere expressly asserts the fact respecting magnetic bodies, or that he made any experiments to confirm it. Yet it may be observed, that in his second memoir on electricity and magnetism† he proved that in a long needle the magnetic force may be conceived to be collected very near each end, which is an indication of the same kind of effect of the theoretical properties of magnetism. It was probably the experimental labours of the English philosophers which led M. Poisson to perform the same office for the magnetic which he had executed so well for the electric theory; to trace the consequences of Coulomb's hypotheses by the aid of powerful and general analytical methods. In February 1824 a memoir of his upon this subject was read to the Institute, and published in 1826, in the memoirs for 1821 and 1822, according to the strange method of publication of the French Academy. In this memoir he obtains expressions for the attractions and repulsions of a body magnetised by influence upon any point, and examines in particular the case in which the body is a sphere. M. Poisson gives the name of magnetic * See his memoir in the Phil. Trans. 1821. † Acad. Par. 1785, p. 578. elements to the small parts of bodies within which the magnetic fluids can be separated. Supposing these elements to be spherical, he would be enabled to determine the conditions of equilibrium of the free magnetic fluid at the surface of each element, by the same analysis as in the case of electricity; using, throughout his researches, those peculiar functions which we have termed Laplace's coefficients, and which introduce such extraordinary facilities into researches of this kind. It further appears, from the nature of the equations (p. 283), that we need not know the form of these elements; for the form of the elements, and the proportion of their sum to the whole mass of the body, enter into the result jointly, so that we do not trace the separate effect of these data. M. Poisson therefore (p. 290) takes the equation of equilibrium on the supposition that the magnetic elements are spherical; and he then finds (p. 306) that this equation coincides with the condition of equilibrium for electricity, on the supposition that the sum of the magnetic elements is equal to the mass of the body (i. e. in his notation, k = 1). And in general (p. 303), the magnetic action of a body of any form is equivalent to that of a thin stratum of magnetic fluid at the surface, although the fluids are separated in every part of the mass. In a subsequent part of his memoir, M. Poisson applies his conclusions to determine the distribution of magnetic fluid in a solid or hollow sphere acted upon by the terrestrial magnetism. He refers to Mr. Barlow's experiments, and to his inference that magnetism resides at the surface alone; he observes that the inference is not warranted, and that the only conclusion which we are justified in drawing by the fact, as compared with the formula, is that the sum of the magnetic elements is equal exactly, or very nearly, to the whole mass of the body. The force exerted by a body in which magnetism is induced, is a joint result of the distribution of the magnetism thus excited, and of the position of the point acted on. The verification of M. Poisson's theory would require experiments made with masses of iron of various forms, as well as measures of the effect on a needle in various situations with reference to the mass; and the theory, thus verified, would disclose to us the distribution of the magnetism at the surface of iron under given circumstances. The verification, with respect to the position of the point acted on, has been executed to a satisfactory extent. M. Poisson observes (p. 336) "that the laws of the deviation of compass needles are in accordance, whether we deduce them from theory, or from observation" as Mr. Barlow had done; "and thus that gentle man's numerous observations are a remarkable confirmation of the theory of magnetism here presented." I do not think it necessary to dwell upon the necessity of a correction for the length of the needle, and its magnetic effect upon the iron sphere, which M. Poisson conceives to be requisite in calculating Mr. Barlow's experiments. But it may be observed, that M. Poisson's assumption that his quantity k (which expresses the ratio of the sum of the magnetic elements of the body, to the sum of all its parts,) cannot exceed unity, does not appear to be incontestable, since it involves the supposition that the whole magnetic attraction or repulsion of each such element is the same as if its form were spherical, which supposition is introduced p. 290 of the memoir. The same volume of the Memoirs of the Institute contains a second memoir of M. Poisson on the same subject, read December 27, 1824. In this the author observes, that though Mr. Barlow's observations afford an important confirmation of the theory, it was desirable that it should be subjected to trials of a more varied kind. On returning to his formulæ, he found that they could be very simply solved for the case of any ellipsoid whatever. Now a very flat ellipsoid may approach indefinitely near to an elliptical or circular plate; a very slender ellipsoid may approach indefinitely to a linear bar. Thus the mathematical theory of certain very obvious and extensive cases was attainable. We do not, however, possess any comparison of experiments with the formulæ thus obtained; and thus the verification of M. Poisson's theory, so far as the distribution of magnetism depends on the form of the mass of iron, is hitherto incomplete. M. Hansteen of Copenhagen, whose valuable work on terrestrial magnetism was published in German in 1819, had inferred from his own experiments, that in a linear magnet the magnetic intensity follows the law of the square of the distance from the middle point more nearly than any other power of that distance, a conclusion different from Coulomb's, as we have seen (p. 11)*. It appears, however, to have been taken for granted, after the verification of the theory by Mr. Barlow's experiments, that it might be considered as established, and that mathematical methods of deduction might for the future be used, not to confirm the truth of its principles, but to apply them to any requisite purposes. The most remarkable and important example of a "Deductio ad Praxin” of this kind, was that which Mr. Barlow made in the * Hansteen, Magnetismus, chap. v. p. 165. case which had first given rise to his researches, the correction of the deviations of ship-compasses produced by the "local attraction"; and it was this case which suggested to M. Poisson some of the problems of his second memoir. Mr. Barlow was soon enabled to perceive, from his own experiments, that the guns and other iron of a vessel produce the same effect as a small sphere of iron in a certain position; and his first idea was to place another iron ball on the opposite side of the compass, so as to counteract this effect. But when a ship moves into various positions, she turns round a vertical axis, which does not coincide with the axis of magnetic position. Therefore the relative magnetic situation of the disturbing and the correcting masses would vary with the changes of position of the vessel; and the correcting ball, in order to discharge its office, must be altered in place or size when the vessel turned its head different ways, an inconvenience which rendered this device almost nugatory. He then proposed to place the ball in a certain fixed position, in which it would double the deviation arising from the local attraction; and finally, when he discovered that the attracting power of iron resided in the surface, he substituted an iron plate for the ball, and thus his apparatus and the mode of using it became convenient and easily managed. The correcting plate so employed would produce the requisite effect if the attraction of the iron in the ship could always be referred to the same virtual centre. The attraction of a mass, however irregular, is equivalent to a single force acting to a single point or "focus of attraction". But this focus may be different in different positions of the irregular mass; for the magnetism which is developed by the earth's action in any mass will depend upon the form and position of its surface; and when the position varies, the position of the resulting attraction with respect to the mass may also vary. Hence, when a ship's compass is disturbed by the action of the irregular mass of iron which the vessel contains, it may happen that the same plate or ball, in the same relative situation, cannot either counteract or double the ship's attraction in all positions. Whether such effects are possible or not must depend upon calculation. By M. Poisson's investigations it appears that this possibility depends on certain conditions, and that it does not exist generally*. But it may be observed that the effect of the attraction of the vessel is greatest (and therefore the necessity of correction greatest) when the dip is considerable, because then the horizontal directive force of terrestrial magnetism is small. Now *Mémoire, 1822, p. 531. in such cases the disturbing masses, which assume their different positions by being turned round a vertical axis, will be nearly in the same magnetical attitude in all their changes, and therefore their effect will not much be altered. Thus Mr. Barlow's correction will be nearly complete in those cases in which it is most important. Mr. Barlow informs me, that in the voyages recently made towards the north pole, advantage was taken of the near coincidence of the magnetic equator with the horizon; and the contrivance of a counteracting plate, which had properly been rejected in other cases for the reasons just mentioned, was adopted with great success. I do not consider it to belong to my present purpose to notice those experimental inquiries concerning magnetism which have not yet been brought into manifest connexion with the theory. One of the most important branches of the subject, that which has to do with Terrestrial Magnetism, has already been the subject of a Report presented to the Association by Prof. Christie, and since published*, and of a supplementary Report by Capt. Sabine in the present volume. I proceed therefore to another of the subjects which are included in my present task. Heat. The doctrine of the Conduction and Radiation of Heat, mathematically treated, is a subject which has excited considerable notice of late years; and its history brings before us several important questions of physics and mathematics. I will speak in order: 1st, of the Experimental Evidence of the Principles of this doctrine; 2ndly, of certain Difficulties which affect the Fundamental Equations; 3rdly, of the Mathematical Processes by which these equations have been treated; and 4thly, of the Application of the Mathematical results to several subjects of speculation. 1. Experimental Thermotical Principles.-The first step in the application of mathematical principles to conducted and radiated heat was made in the Principia. "It was in the destiny of that great work," says Fourier, "to exhibit, or at least to indicate, the causes of the principal phænomena of the universe." Newton assumed, as a simple rule evidently agreeable to facts, that the rate at which a body parts with its heat is proportional to the excess of heat; and on this assumption he rested the verification of his scale of temperatures. It is an easy deduction from this law, that if times of cooling be taken in arithmetical progression, the heat will decrease in geometrical progression. Kraft, and after him Richman, tried to verify this law by direct experiments on the cooling of vessels of warm water; and from 1835. * Report of Third Meeting, p. 105. C |