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to the analytical subject which we have considered. This question connected with the theory of radiant heat has just been discussed by the illustrious author of the Mécanique céleste, to whom all the chief branches of mathematical analysis owe important discoveries. (Connaissance des Temps, years 1824-5.)

The new theories explained in our work are united for ever to the mathematical sciences, and rest like them on invariable foundations; all the elements which they at present possess they will preserve, and will continually acquire greater extent. Instruments will be perfected and experiments multiplied. The analysis which we have formed will be deduced from more general, that is to say, more simple and more fertile methods common to many classes of phenomena. For all substances, solid or liquid, for vapours and permanent gases, determinations will be made of all the specific qualities relating to heat, and of the variations of the coefficients which express them'. At different stations on the earth observations will be made, of the temperatures of the ground at different depths, of the intensity of the solar heat and its effects, constant or variable, in the atmosphere, in the ocean and in lakes; and the constant temperature of the heavens proper to the planetary regions will become known. The theory itself

1 Mémoires de l'Académie des Sciences, Tome VIII., Paris 1829, contain on pp. 581-622, Mémoire sur la Théorie Analytique de la Chaleur, par M. Fourier. This was published whilst the author was Perpetual Secretary to the Academy. The first only of four parts of the memoir is printed. The contents of all are stated. I. Determines the temperature at any point of a prism whose terminal temperatures are functions of the time, the initial temperature at any point being a function of its distance from one end. II. Examines the chief consequences of the general solution, and applies it to two distinct cases, according as the temperatures of the ends of the heated prism are periodic or not. III. Is historical, cnumerates the earlier experimental and analytical researches of other writers relative to the theory of heat; considers the nature of the transcendental equations appearing in the theory; remarks on the employment of arbitrary functions; replies to the objections of M. Poisson; adds some remarks on a problem of the motion of waves. IV. Extends the application of the theory of heat by taking account, in the analysis, of variations in the specific coefficients which measure the capacity of substances for heat, the permeability of solids, and the penetrability of their surfaces. [A. F.]

2 Mémoires de l'Académie des Sciences, Tome VII., Paris, 1827, contain on pp. 569–604, Mémoire sur les températures du globe terrestre et des espaces planétaires, par M. Fourier. The memoir is entirely descriptive; it was read before the Academy, 20 and 29 Sep. 1824 (Annales de Chimie et de Physique, 1824, xxvII. p. 136). [A. F.]

will direct all these measures, and assign their precision. No considerable progress can hereafter be made which is not founded on experiments such as these; for mathematical analysis can deduce from general and simple phenomena the expression of the laws of nature; but the special application of these laws to very complex effects demands a long series of exact observations.

The complete list of the Articles on Heat, published by M. Fourier, in the Annales de Chimie et de Physique, Series 2, is as follows:

1816. III. pp. 350-375. Théorie de la Chaleur (Extrait). Description by the author of the 4to volume afterwards published in 1822 without the chapters on radiant heat, solar heat as it affects the earth, the comparison of analysis with experiment, and the history of the rise and progress of the theory of heat.

1817. IV. pp. 128-145. Note sur la Chaleur rayonnante. Mathematical sketch on the sine law of emission of heat from a surface. Proves the author's paradox on the hypothesis of equal intensity of emission in all directions.

1817. VI. pp. 259-303. Questions sur la théorie physique de la chaleur rayonnante. An elegant physical treatise on the discoveries of Newton, Pictet, Wells, Wollaston, Leslie and Prevost.

1820. XIII. pp. 418–438. Sur le refroidissement séculaire de la terre (Extrait). Sketch of a memoir, mathematical and descriptive, on the waste of the earth's initial heat.

1824. XXVII. pp. 136-167. Remarques générales sur les températures du globe terrestre et des espaces planétaires. This is the descriptive memoir referred to above, Mém. Acad. d. Sc. Tome VII.

1824. XXVII. pp. 236-281. Résumé théorique des propriétés de la chaleur rayonnante. Elementary analytical account of surface-emission and absorption based on the principle of equilibrium of temperature.

1825. XXVIII. pp. 337-365. Remarques sur la théorie mathématique de la chaleur rayonnante. Elementary analysis of emission, absorption and reflection by walls of enclosure uniformly heated. At p. 364, M. Fourier promises a Théorie physique de la chaleur to contain the applications of the Théorie Analytique omitted from the work published in 1822.

1828. XXXVII. pp. 291-315. Recherches expérimentales sur la faculté conductrice des corps minces soumis à l'action de la chaleur, et description d'un nouveau thermomètre de contact. A thermoscope of contact intended for lecture demonstrations is also described. M. Emile Verdet in his Conférences de Physique, Paris, 1872. Part I. p. 22, has. stated the practical reasons against relying on the theoretical indications of the thermometer of contact. [A. F.]

Of the three notices of memoirs by M. Fourier, contained in the Bulletin des Sciences par la Société Philomatique, and quoted here at pages 9 and 11, the first was written by M. Poisson, the mathematical editor of the Bulletin, the other two by M. Fourier. [A. F.]

THEORY OF HEAT.

Et ignem regunt numeri.-PLATO1.

CHAPTER I.

INTRODUCTION.

FIRST SECTION.

Statement of the Object of the Work.

1. THE effects of heat are subject to constant laws which cannot be discovered without the aid of mathematical analysis. The object of the theory which we are about to explain is to demonstrate these laws; it reduces all physical researches on the propagation of heat, to problems of the integral calculus whose elements are given by experiment. No subject has more extensive relations with the progress of industry and the natural sciences; for the action of heat is always present, it penetrates all bodies and spaces, it influences the processes of the arts, and occurs in all the phenomena of the universe.

When heat is unequally distributed among the different parts of a solid mass, it tends to attain equilibrium, and passes slowly from the parts which are more heated to those which are less; and at the same time it is dissipated at the surface, and lost in the medium or in the void. The tendency to uniform distribution and the spontaneous emission which acts at the surface of bodies, change continually the temperature at their different points. The problem of the propagation of heat consists in

1 Cf. Plato, Timæus, 53, B.

ὅτε δ' ἐπεχειρεῖτο κοσμεῖσθαι τὸ πᾶν, πῦρ πρῶτον καὶ γῆν καὶ ἀέρα καὶ ὕδωρ........ διεσχηματίσατο [ὁ θεὸς] εἴδεσί τε καὶ ἀριθμοῖς. [Α. F.]

determining what is the temperature at each point of a body at a given instant, supposing that the initial temperatures are known. The following examples will more clearly make known the nature of these problems.

2. If we expose to the continued and uniform action of a source of heat, the same part of a metallic ring, whose diameter is large, the molecules nearest to the source will be first heated, and, after a certain time, every point of the solid will have acquired very nearly the highest temperature which it can attain. This limit or greatest temperature is not the same at different points; it becomes less and less according as they become more distant from that point at which the source of heat is directly applied.

When the temperatures have become permanent, the source of heat supplies, at each instant, a quantity of heat which exactly compensates for that which is dissipated at all the points of the external surface of the ring.

If now the source be suppressed, heat will continue to be propagated in the interior of the solid, but that which is lost in the medium or the void, will no longer be compensated as formerly by the supply from the source, so that all the temperatures will vary and diminish incessantly until they have become equal to the temperatures of the surrounding medium.

3. Whilst the temperatures are permanent and the source remains, if at every point of the mean circumference of the ring an ordinate be raised perpendicular to the plane of the ring, whose length is proportional to the fixed temperature at that point, the curved line which passes through the ends of these ordinates will represent the permanent state of the temperatures, and it is very easy to determine by analysis the nature of this line. It is to be remarked that the thickness of the ring is supposed to be sufficiently small for the temperature to be sensibly equal at all points of the same section perpendicular to the mean circumference. When the source is removed, the line which bounds the ordinates proportional to the temperatures at the different points will change its form continually. The problem consists in expressing, by one equation, the variable

form of this curve, and in thus including in a single formula all the successive states of the solid.

4. Let z be the constant temperature at a point m of the mean circumference, a the distance of this point from the source, that is to say the length of the arc of the mean circumference, included between the point m and the point o which corresponds to the position of the source; z is the highest temperature which the point m can attain by virtue of the constant action of the source, and this permanent temperature z is a function f(x) of the distance x. The first part of the problem consists in determining the function f(x) which represents the permanent state of the solid.

Consider next the variable state which succeeds to the former state as soon as the source has been removed; denote by t the time which has passed since the suppression of the source, and by the value of the temperature at the point m after the time t. The quantity v will be a certain function F (x, t) of the distance x and the time t; the object of the problem is to discover this function F(x, t), of which we only know as yet that the initial value is ƒ (x), so that we ought to have the equation f(x) = F (x, 0).

5. If we place a solid homogeneous mass, having the form of a sphere or cube, in a medium maintained at a constant temperature, and if it remains immersed for a very long time, it will acquire at all its points a temperature differing very little from that of the fluid. Suppose the mass to be withdrawn in order to transfer it to a cooler medium, heat will begin to be dissipated at its surface; the temperatures at different points of the mass will not be sensibly the same, and if we suppose it divided into an infinity of layers by surfaces parallel to its external surface, each of those layers will transmit, at each instant, a certain quantity of heat to the layer which surrounds it. If it be imagined that each molecule carries a separate thermometer, which indicates its temperature at every instant, the state of the solid will from time to time be represented by the variable system of all these thermometric heights. It is required to express the successive states by analytical formulæ, so that we

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