Scientific Memoirs/1/On the Mathematical Theory of Heat
Article VI.
On the Mathematical Theory of Heat; by S. D. Poisson, Member of the Institute, &c [1]
From the Annales de Chimie et de Physique, vol. lix. p. 71 et seq.
The work which I have just published under the title of The Mathematical Theory of Heat (Théorie Mathematique de la Chaleur), forms the second part of a treatise on Mathematical Physics (Physique Mathématique), the first of which is the New Theory of Capillary Action (Nouvelle Théorie de l'Action Capillaire), which appeared four years ago. It contains twelve chapters, preceded by some pages in which I recapitulate in a few words the first applications of the calculus which have been made to the theory of heat, and the principal researches of geometers upon that subject, which have been made of late years, namely, since the first Memoir presented by Fourier to the Institute in 1807. I will here transcribe the contents of the Preface, on the important question of the heat of the earth.
"In applying to the earth the general solution of the problem of a sphere at first heated in any manner whatever, Laplace was led to participate in the opinion of Fourier, which attributes to the primitive heat of the earth the increase in temperature which is observed in descending from the surface, and the amount of which is not the same in all localities. This hypothesis of a temperature proceeding from the original heat of the globe (la chaleur d'origine), and which must rise to millions of degrees in its central layers, has been generally adopted; but the difficulties it presents appear to me to render it improbable. I have proposed a different explanation of the increasing temperature which has long since been observed at all depths to which man has penetrated.
"According to this new explanation the phænomenon depends on the inequality of temperature of those regions of space which the earth successively passes through in its translatory motion, and which are common to the sun and all the planets. It would be indeed opposed to all probability that the temperature of space should everywhere be the same; the variations to which it is subject from one point to another, separated by very great distances, may be very considerable, and ought to produce corresponding variations in the temperature of the earth, extending to various depths according to their duration and amplitude. Suppose, for the sake of example, a block of stone transported from the equator to our latitudes; its cooling will have commenced at the surface, and then become propagated into the interior; and if the cooling has extended throughout the whole mass, because the time of its transportation has been very short, that body thus transported to our climate will present the phænomenon of an increase of temperature with the distance from the surface. The earth is in the case of this block of stone;—it is a body coming from a region the temperature of which was higher than that of the place in which it now is; or we may regard it as a thermometer moveable in space, but which has not had time, on account of its magnitude and according to its degree of conducting power, to take throughout its mass the temperatures of the different regions through which it has passed. At present the degree of temperature of the globe is increasing below the surface; the contrary has in former times been, and will hereafter be, the case: besides, at epochs separated by many series of ages this temperature must have been, and will in future be, much higher or lower than what it is at present; a circumstance, which renders it impossible that the earth should always be habitable by man, and has perhaps contributed to the successive revolutions the traces of which have been discovered in its exterior crust. It is necessary to observe that the alternations of temperature of space are positive causes which have an increasing influence upon the heat of the globe at least near its surface; while the original heat of the earth (chaleur d'origine de la terre), however slow it may be in dissipating, is but a transitoiy circumstance, the existence of which it would not be possible at the present epoch to demonstrate, and to which we should not be forced to have recourse as a hypothesis except in the case of the permanent and necessary causes being insufficient to explain the different phænomena."
The following are the titles of the different chapters of the work, together with a short abstract of the contents of each.
Chapter I. Preliminary Notions.—After having given the definition of temperature and many other definitions, it is explained how we have been led to the principle of a continual radiation and absorption of heat by the molecules of all bodies. The interchange of heat between material particles of an insensible magnitude, but yet comprising immense numbers of molecules, cannot disturb the equality of their temperatures when it actually exists. From this condition we conclude, that for each particle the ratio of the emitting to the absorbing power is independent of the substance and of density, and that it can only depend on temperature. In the case of the inequality of temperatures, we give the general expression of their variations during every instant, equal and contrary for two material particles, radiating one toward the other. We also give the law of absorption of radiant heat in the interior of homogeneous bodies.
Chapter II. Laws of Radiant Heat.—If a body be placed within a vacuous sphere on every side (enceinte vide fermée de toutes parts), the temperature of which is supposed to be invariable and everywhere the same, we demonstrate that the result of the interchange of heat between an element of its surface and an element of the surface of the inclosing sphere, is independent of the matter of which the sphere is formed, and proportional, cæteris paribus, to the cosines of the angle which the normal to the second element forms with the right line from one to the other element. Experiments, not as yet made, only can decide whether this law of the cosine is equally applicable to the elements of the surface of the body, of which the temperature is not invariable like that of the sphere; and until such experiments are made we may be allowed to doubt its existence while the body is heating or cooling. By considering the number of successive reflexions which take place at the surface of the sphere we demonstrate also that in general the passage (flux) of heat through every element in the surface of the body which it contains is independent of the form, of the dimensions, and of the material of the sphere; there is no exception, but when the heat, in the series of reflexions which it experiences, falls one or many times upon the surface of the body. It follows from this theorem that a thermometer placed in any point whatever of the space which the sphere terminates, will finally indicate the same temperature, which will be equal to that of the sphere; but in the case of the exception just mentioned, the time which it will employ in attaining that temperature will vary according to the place it occupies. The general expression of the passage of heat through every element of the surface of a body of which the temperature varies, is composed of one factor relative both to the state of that surface and to the material of the body, multiplied by the difference of two similar functions, one of which depends on the variable temperature of the body, the other on the fixed temperature of the sphere, which are the same for all bodies; a result which agrees with the law of cooling in vacuo discovered by MM. Dulong and Petit. We next suppose in this second chapter, that many bodies differing in temperature are contained in the sphere of which the temperature is constant, and arrive then at a general formula, which will serve to solve the problems of the catoptrics of heat, the principal applications of which we indicate. When all these bodies form round one of them a closed sphere the temperature of which, variable with the time, is not the same throughout, the passage of heat to the surface of the interior body does not depend on its temperature and that of the inclosure only, at least when these bodies are formed of the same material. After having considered the influence of the air upon radiation which we had at first eliminated, we give at the end of this chapter a formula which expresses the instantaneous variations of temperature of two material particles of insensible magnitude, by means of which the exchange of heat takes place after one or many reflexions upon the surfaces of other bodies through air or through any gas whatever.
Chapter III. The Laws of Cooling in Bodies having the same Temperature throughout.—While a homogeneous body of small dimensions is heating or cooling, its variable temperature is the same at every point; but if the body is composed of many parts formed of different substances in juxtaposition, they may preserve unequal temperatures during all the time that these temperatures vary, as we show in another chapter. In the present we determine, in functions of the time, the velocity and the temperature which we suppose to be common to all the points in a body placed alone in a sphere either vacuous or full of air, and the temperature of which is variable. If the sphere contains many bodies subject to their mutual influence upon each other, the determination of their temperatures would depend on the integration of a system of simultaneous equations, which are only linear in the case of ordinary temperatures, but in which we cannot separate the variables when we investigate high temperatures, and when the radiation is supposed not to be proportional to their differences.
Experiment has shown that in a cooling body, covered by a thin layer or stratum of a substance different from that of which it is itself composed, the velocity of refrigeration only arrives at its maximum when the thickness of this additive stratum, though always very small, has notwithstanding attained a certain limit. We develop the consequences of this important fact in what regards extension of molecular radiation, and explain how those consequences agree with the expression of the passage of heat found in the preceding chapter.
Chapter IV. Motion of Heat in the Interior of Solid or Liquid Bodies.—We arrive by two different processes at the general equation of the motion of heat; these two methods are exempt from the difficulties which the Committee of the Institute, which awarded the prize of 1812[2] to Fourier, had raised against the exactitude of the principle upon which his method was sustained. The equation under consideration is applicable both to homogeneous and heterogeneous bodies, solid or fluid, at rest or in motion. It was unnecessary, as they appeared to have thought, to find for fluids an equation different from the one I obtained long since for heterogeneous bodies. The variations of temperature which take place at every instant, and arise from the mutual radiation of the neighbouring molecules, depend in fact only on their actual positions, and not at all on the positions in which they will be the instant after in consequence of the motions produced by their calorific action or by other causes: it is thus that in the problem of the flux and reflux of the tides, for example, we calculate the attraction of the sea upon each point of its mass, as if it were solid and at rest at the moment under consideration, and independently of the motions which this attraction may produce.
Notwithstanding that the interior radiation takes place only between molecules the temperatures of which are extremely different, the equation of motion of the heat contains terms derived from the squares of their differences, and of the same order of magnitude as those which result from their first power; so that the exact equation differs, in the case of a homogeneous body, from that which we had already given; and it is not, like that, independent of the conductibility when the body has arrived at an invariable state. This equation of partial differences changes its form, when we cannot consider the extent of the interior radiation as insensible; it is then of a higher order, which introduces, in its integral, new constants or arbitrary functions. From this a difficulty of analysis arises, of which we give the solution, and explain how in every case the redundant quantities will be made to disappear, as will be seen from a particular example in another chapter. We form in this the general expression of the passage of heat through every element of a surface traced in the interior of a body which is heated or cooled, or has arrived at an invariable state, and in which the extent of the interior radiation is considered as insensible. This passage proceeds from the exchange of heat between the molecules of the two parts of that body near their surface of separation, and the temperatures of which are very different; whilst the interior passage results from the exchanges between the molecules adjacent to the surface of the body and those of a surrounding medium, or of other bodies which may have much higher or much lower temperatures; and notwithstanding that the respective magnitudes of these two passages (ces deux flux), due to causes also unequal, must be of the same order and comparable with one another. We show how that condition is fulfilled, by means of a quantity resulting from the rapid decrease of temperature which takes place very near the surface of a body whilst heating or cooling. In this manner interior and exterior passages are found united with one another; and the law of interior conductibility expressed in functions of the temperature is deduced from that of exterior radiation which MM. Dulong and Petit have discovered.
In a homogeneous prism which has arrived at an invariable state, and the lateral surface of which is supposed to be impermeable to heat and its two bases retained at constant temperatures, the passage of heat across every section perpendicular to its length is the same throughout its length. Its magnitude is proportional to the temperature of the two bases, and in the inverse ratio of the distance which separates them. This principle is easy to demonstrate, or rather it may be considered as evident. Thus expressed, it is independent of the mode of communication of heat, and it takes place whatever be the length of the prism: but it was erroneous to have attributed it without restriction to the infinitely thin slices of one body, the temperature of which varies, either with the time, or from one point to another; and to have excluded from it the circumstance, that the equation of the movement of heat, deduced from that of extension, is independent of any hypothesis and comparable in its generality to the theorems of statics. When we make no supposition respecting the mode of communication of heat, or the law of interior radiation, the passage of heat through each face of an infinitely thin slice is no longer simply proportional to the infinitely small difference of the temperatures of the two faces, or in the inverse ratio of the thickness of the slices; the exact expression of it will be found in the chapter in which we treat specially of the distribution of heat in a prismatic bar.
Chapter V. On the Movement of Heat at the Surface of a Body of any Form.—We demonstrate that the passages of heat are equal, or become so after a very short time, in the two extremities of a prism which has for its base an element of the surface of a body, and is in height a little greater than the thickness of the superficial layer, in which the temperature varies very rapidly. From this equality, and from the expression of the exterior radiation, given by observation, we determine the equation of the motion of heat at the surface of a body of any form whatsoever. The expression of the interior passage not being applicable to the surface itself, it follows that the demonstration of this general equation, which consists in immediately equalizing that expression to the expression of the exterior radiation, is altogether illusory.
When a body is composed of two parts of different materials, two equations of the motion of heat exist at their surface of separation, which are demonstrated in the same manner as the equation relative to the surface; they contain one quantity depending on the material of those two parts respectively, and which can only be determined by experiment.
Chapter VI. A Digression on the Integrals of Equations of partial Differences.—By the consideration of series, we demonstrate that the number of arbitrary constants contained in the complete integral of a differential equation ought always to be equal to that which indicates the order of that equation: we prove by the same means, that in the integral of an equation of partial differences the number of arbitrary functions may be less, and change as the integral is developed in series, according to the powers of one or other variable; and when the equation of partial differences is linear, we show that by conveniently choosing this variable all the arbitrary functions may disappear and be replaced by constants, infinite in number, without the integral ceasing to be complete. To elucidate these general considerations, we apply them to examples by means of which we show that the different integrals, in the series of the same equation of partial differences, are transformed into one another, and may be expressed under a finite form by definite integrals, which also contain one or several arbitrary functions. In the single case, in which the integral in series contains only arbitrary constants, every term of the series by itself satisfies the given equation, so that the general integral is found expressed by the sum of an infinite number of particular integrals. Integrals of this form have appeared from the origin of the calculus of partial differences; but in order that their use in different problems should not leave any doubt respecting the generality of the solutions, it would have been necessary to have demonstrated à priori, as I did long since, that these expressions in series, although not containing any arbitrary function, as well as those containing a greater or smaller number of them, are not less on that account the most general solutions of equations of partial differences; or else it would have been necessary to verify in every example that, after having satisfied all the equations of one problem relative to contiguous points infinite in number, the series of this nature might still represent the initial and entirely arbitrary state of this system of material points; a verification which, until now, it has not been possible to give, except in very particular cases. The solution which Fourier was the first to offer of the problem of the distribution of heat in a homogeneous sphere, of which all the points equidistant from the centre have equal temperatures, does not satisfy for example either of these two conditions; it was no doubt on this account that the members of the Committee, whose judgement we mentioned above, thought that his (Fourier's) analysis was not satisfactory in regard to generality; and, in fact, in this solution it is not at all demonstrated that the series which expresses the initial temperature can represent a function, entirely arbitrary, of the distance from the centre.
For the use of these series of particular solutions, it will be necessary to proceed in a manner proper to determine their coefficients according to the initial state of the system. On the occasion of a problem relative to the heat of a sphere composed of two different substances, I have given for this purpose in the Journal de l'Ecole Polytechnique, (cahier 19, p. 377 et seq.,) a direct and general method, of which I have since made a great number of applications, and which I shall also constantly follow in this work. The Sixth Chapter contains already the application to the general equations of the motion of heat in the interior and on the surface of a body of any form either homogeneous or heterogeneous. It leads in every case to two remarkable equations, one of which serves to determine, independently of one another, the coefficients of the terms of each series, and the other to demonstrate the reality of the constant quantities by which the time is multiplied in all these terms. These constants are roots of transcendental equations, the nature of which it will be very difficult to discover, by reason of the very complicated form of these equations. From their reality this general consequence is drawn; viz. when a body, heated in any manner whatever, is placed in a medium the temperature of which is zero, it always attains, before its complete cooling, a regular state in which the temperatures of all its points decrease in the same geometrical progression for equal increments of time. We shall demonstrate in another chapter, that, if that body is a homogeneous sphere, these temperatures will be equal for all the points at an equal distance from the centre, and the same as if the initial heat of each of its concentric strata had been uniformly distributed throughout its extent.
The equations of partial differences upon which depend the laws of cooling in bodies are of the first order in regard to time, whilst the equations relative to the vibrations of elastic bodies and of fluids are of the second order; there result essential differences between the expressions of the temperatures and those of the velocities at a given instant, and for that reason it appears at least very difficult to conceive that the phænomena which may result from a molecular radiation should be equally explicable by attributing them to the vibrations of an elastic fluid. When we have obtained the expressions of the unknown quantities in functions of the time, in either of these kinds of questions, if we make the time in them equal to zero, we deduce from that, series of different forms which represent, for all the points of the system which we consider, arbitrary functions, continuous or discontinuous, of their coordinates. These expressions in series, although we might not be able to verify them, except in particular examples, ought always to be admitted as a necessary consequence of the solution of every problem, the generality of which has been demonstrated à priori; it will however be desirable that we should also obtain them in a more direct manner; and we might perhaps so attain them, by means of the analysis of which I had made use in my first Memoir on the theory of heat, to determine the law of temperatures in a bar of a given length, according to the integral under a finite form of the equation of partial differences.
Chapter VII. A Digression on the Manner of expressing Arbitrary Functions by Series of Periodical Quantities.—Lagrange was the first to give a series of quantities proper to represent the values of an arbitrary function, continuous or discontinuous, in a determined interval of the values of the variable. This formula supposes that the function vanishes at the two extremes of this interval; it proceeds according to the sines of the multiples of the variable; many others exist of the same nature which proceed according to the sines or cosines of these multiples, even or uneven, and which differ from one another in conditions relative to each extreme. A complete theory of formulæ of this kind will be found in this chapter, which I have abstracted from my old memoirs, and in which I have considered the periodical series which they contain as limits of other converging series, the sums of which are integrals, themselves having for limits the arbitrary functions which it is our object to represent. Supposing in one or other of these expressions in series, the interval of the values of the variable for which it takes place to be infinite, there results from it the formula with a double integral, which belongs to Fourier; it is extended without difficulty, as well as each of those which only subsists for a limited interval, to two or a greater number of variables.
Chapter VIII. Continuation of the Digression on the Manner of representing Arbitrary Functions by Series of Periodical Quantities.—An arbitrary function of two angles, one of which is comprised between zero and 180°, and the other between zero and 360°, may always be represented between those limits by a series of certain periodical quantities, which have not received particular denominations, although they have special and very remarkable properties. It is to that expression in series that we have recourse in a great number of questions of celestial mechanics and of physics, relative to spheroids; it had however been disputed whether they agreed with any function whatever; but the demonstration of this important formula, which I had already given and now reproduce in this chapter, will leave no doubt of its nature and generality. This demonstration is founded on a theorem, which is deduced from considerations similar to those of the preceding chapter. We examine what the series becomes at the limits of the values of the two angles; we then demonstrate the properties of the functions of which its terms are formed; then it is shown that they always end by decreasing indefinitely, which is a necessary consequence and sufficient to prevent the series from becoming diverging, for which purpose its use is always allowable. Finally, it is proved, that for the same function there is never more than one development of that kind; which does not happen in the developments in series of sines and cosines of the multiples of the variables. This chapter terminates with the demonstration of another theorem, by means of which we reduce a numerous class of double integrals to simple integrals.
Chapter IX. Distribution of Heat in a Bar, the transverse Dimensions of which are very small.—We form directly the equation of the motion of heat in a bar, either straight or curved, homogeneous or heterogeneous, the transverse sections of which are variable or invariable, and which radiates across its lateral surface. We then verify the coincidence of this equation with that which is deduced from the general equation of Chapter IV., when the lateral radiation is abstracted and the bar is cylindrical or prismatic. This equation is first applied to the invariable state of a bar the two extremities of which are kept at constant and given temperatures. It is then supposed, successively, that the extent of the interior radiation is not insensible, that the exterior radiation ceases to be proportional to the differences of temperature, that the exterior conductibility varies according to the degree of heat, and the influence of those different causes on the law of the permanent temperatures of the bar is determined. Formulæ are also given, which will serve to deduce from this law, by experiment, the respective conductibility of different substances, and the quantity relative to the passage from one substance into another, in the case of a bar formed of two heterogeneous parts placed contiguous to and following one another. After having thus considered in detail the case of permanent temperatures, we resolve the equation of partial differences relative to the case of variable temperatures; which leads to an expression of the unknown quantity of the problem, in a series of exponentials, the coefficients of which are determined by the general process indicated in Chapter VII., whatever may be the variations of substance and of the transverse sections of the bar. We then apply that solution to the principal particular cases. When the bar is indefinitely lengthened, or supposed to be heated only in one part of its length, the laws of the propagation of heat on each side of the heated place are determined; this propagation is instantaneous to any distance; a result of the theory presenting a real difficulty, but the explanation of which is given.
Chapter X. On the Distribution of Heat in Spherical Bodies.—The problem of the distribution of heat in a sphere, all the points of which equally distant from the centre have equal temperatures, is easily brought to a particular case of the same question with regard to a cylindrical bar. It is also solved directly; the solution is then applied to the two extreme cases, one of a very small radius, and another of a very great one. In the case of an infinite radius, the laws are inferred of the propagation of caloric in a homogeneous body, round the part of its mass to which the heat has been communicated, similarly in all directions.
We then determine the distribution of heat in a homogeneous sphere covered with a stratum, also homogeneous, formed of a substance different from that of the nucleus. During the whole time of cooling, the temperature of this stratum, however small its thickness may be, is different from that of the sphere in the centre, and the ratio of the temperatures of these two parts, at the same instant, depends on the quantity relative to the passage from one substance into the other, of which we have already spoken. From this circumstance an objection arises against the method employed by all natural philosophers to determine, by the comparison of the velocities of cooling, the ratio of the specific heat of different bodies, after having brought their surfaces to the same state by means of a very thin stratum of the same substance for all these bodies. The quantity relative to the passage of the heat of each body in the additive stratum, is contained in the ratio of the velocities of cooling; it is therefore necessary that it should be known in order to be able to deduce from this ratio, that of their specific heats. A recent experiment by M. Melloni proves that a liquid contained in a thin envelope, the interior surface of which is successively placed in different states by polishing or scratching it, always cools with the same velocity, whilst the ratios of the velocity change very considerably, as was known long before, when it is the exterior part of the vessel that is more or less polished or scratched. The quantity relative to the passage of caloric across the surface of separation of the vessel and the liquid, is therefore independent of the state of that surface, a circumstance which assimilates the cooling power of liquids to that of the stratum of air in contact with bodies, which in the same manner does not depend on the state of their surface, according to the experiments of MM. Dulong and Petit.
When a homogeneous sphere, the cooling of which we are considering, is changed into a body terminated by an indefinite plane, and is indefinitely prolonged on one side only of that plane, the analytical expression for the temperature of any point whatever changes its form, in such a manner that that temperature, instead of tending to diminish in geometrical progression, converges continually towards a very different law, which depends on the initial state of the body; but however great a body may be, it has always finite and determined dimensions; and it is always the law of final decrease enunciated in Chapter VI. which it is necessary to apply; even in the case, for example, of the cooling of the earth.
If the distribution of heat in a sphere, or in a body of another form, has been determined, by supposing this body to be placed in a medium the temperature of which is zero, this first solution of the problem may afterwards be extended to the case in which the exterior temperature varies according to any law whatever. In my first Memoir on the theory of heat, I have followed, in regard to this part of the question, a direct method applicable to all cases. According to this method, one part of the value of the temperature in a function of the time is expressed in the general case by a quadruple integral, which can always be reduced to a double integral like each of the other parts. By the method which I have used to effect this reduction we obtain the value of different definite integrals, which it would be difficult in general to determine in a different manner, and the accuracy of which is verified whenever they enter into known formulæ.
Chapter XI. On the Distribution of Heat in certain Bodies, and especially in a homogeneous Sphere primitively heated in any Manner—It is explained how, in every case, the complete expression of exterior temperature, which may depend on the different sources of heat, and which must be employed in the equation of the motion of heat relative to the surface of bodies submitted to their influence, will be formed.
After having enumerated the different forms of bodies for which we have hitherto arrived at the solution of the problem of the distribution of heat, the complete solution is given for the case of a homogeneous rectangular parallelopiped the six faces of which radiate unequally.
In order to apply the general equations of the fourth and fifth chapters to the case of a homogeneous sphere primitively heated in any manner, the orthogonal coordinates in them are transformed into polar coordinates; the temperature at any instant and in any point is then expressed by means of the general series of Chapter VIII., and of the integrals found in Chapter VI.; the coefficients of that series are next determined according to the initial state of the sphere, by supposing at first the exterior temperature to be zero: by the process already employed in the preceding Chapter, this solution is finally extended to the case of an exterior temperature, varying with the time and from one point to another. Among the consequences of this general solution of the problem the most important is that for which we are indebted to Laplace; it consists in this: That in a sphere of very large dimensions, and at distances from the surface very small in proportion to its radius, the part of the temperature independent of the time does not vary sensibly with these distances; and, that upon the normal at each point, whether at the surface or at an inconsiderable depth, it may be regarded as equal to the invariable part of the exterior temperature which corresponds to the same point. Hence it results, that the increase of heat in the direction of the depth which is observed near the surface of the earth cannot be attributed to the inequality of temperatures of different climates, and that it is necessary to look for the cause in circumstances which vary very slowly with the time. Whatever this cause may be, the difference of the mean temperatures of the surface and beyond, corresponding to the same point of the superficies, is proportional (according to a remark made by Fourier) to the increase of temperature upon the normal referred to the unity of length, so that this difference may be determined from the observed increase, and from a quantity depending on the nature of the ground. This remark and that of Laplace are not applicable to the localities where the temperature varies very rapidly round the vertical: it is shown that in these cases of exception the temperature varies even upon the vertical: and the law of this variation is determined from the variation which has taken place at the surface or in the exterior temperature. The mean temperature at a very small distance contains also a term which is not proportional to this depth, and which arises from the influence of the heat on the conductibility of the substance.
Chapter XII. On the Motion of Heat in the Interior and upon the Surface of the Earth.—It is shown that the formulæ of the preceding chapter, although relating to a homogeneous sphere the surface of which is everywhere in the same state, may notwithstanding serve to determine the temperatures of the points of the earth at a distance from the surface which is very small with regard to its radius, but which exceeds however all accessible depths. These formulæ contain two constants, depending on the nature of the soil, the numerical values of which may be determined in every point of the globe from the temperatures observed at different known depths.
Observation in harmony with theory shows that the diurnal inequalities of the temperature of the earth disappear at very small depths, and the annual inequalities at greater depths, in such a manner that at a distance from the surface of about 20 metres and beyond those two kinds of inequalities are entirely insensible. In this chapter are given tables of the temperatures, indicated by the thermometer, of the caves of the Observatory, at the depth of 28 metres. The mean of 352 observations, made from 1817 to the end of 1834, is 11°·834.
The increase of the mean temperature of the earth, in proportion as we descend below the surface, has long been established as a fact in all deep places, at different latitudes, and at different elevations of the soil above the level of the sea. The most adequate means to determine it is by sounding and boring. The results, still very few, which have hitherto been obtained are given. At Paris, this increase appears to be one degree for about 38 metres of increase in depth.
As to the cause of this phænomenon, the difficulties are stated which the explanation of Fourier presents, founded upon the original heat of the globe, still sensible at the present time near the surface; the new explanation alluded to at the beginning of this article is then proposed. The following reflections extracted from the work tend to prove that the solidification of the earth must have commenced by central strata, and that before reaching the surface the cooling of the globe must have been incomparably more rapid.
"The nearly spherical form of the earth and planets, and their flattening at the poles of rotation, evidently show that these bodies were originally in a fluid or perhaps in an aëriform state. Beginning from this initial state, the earth could not, wholly or partly, become solid, except by a loss of heat arising from its temperature exceeding that of the medium in which it was placed. But it is not demonstrated that the solidification of the earth could have commenced at the surface and been propagated towards the centre, as the state of the globe still fluid in the greatest part of the interior would lead us to suppose; the contrary' appears to me more probable. For the extreme parts, or those nearer to the surface, being the first cooled, must have descended to the interior and been replaced by internal portions which had ascended to cool at the surface and to descend again in their turn. This double current must have maintained an equality of temperature in the mass, or at least must have prevented the inequality from becoming in any way so great as in a solid body, which cools from the surface; and we may add that this mixture of the parts of the fluid, and the equalization of their temperatures, must have been favoured by the oscillations of the whole mass, which must have taken place until the globe attained a permanent figure and rotation. On the other hand, the excessively great pressure sustained by the central strata may have determined their solidification long before that of those nearer the surface; that is to say, the first may have become solid by the effect of this extreme pressure at a temperature equal or even superior to that of the strata more distant from the centre, and consequently subjected to a much less degree of pressure. Experiment has shown, for example, that water at the ordinary temperature being submitted to a pressure of 1000 atmospheres, experiences a condensation of about 120th of its primitive volume. Now let us conceive a column of water whose height is equal to one radius of the globe, and let us reduce its weight to half of that which we observe at the surface of the earth, in order to render it equal to the mean gravity which would exist along each radius of the earth upon the hypothesis of its homogeneity; the inferior strata of this liquid column would experience a pressure of more than three millions of atmospheres, or equal to more than three thousand times the pressure which would reduce water to 1920ths of its volume; but without knowing the law of the compression of this liquid, and although we do not know in what manner this law may depend on the temperature, we may believe, notwithstanding, that so enormous a pressure would reduce the inferior strata of the mass of water to the solid state, even when the temperature is very high. It seems therefore more natural to conceive that the solidification of the earth began at the centre and was successively propagated towards the surface: at a certain temperature, which might be extremely high, the strata nearer the centre became at first solid, by reason of the excessive pressure which they experienced ; the succeeding strata were then solidified at a lower temperature and under a less degree of pressure, and thus in progressive succession to the surface."
If the increase observed in the temperature of the earth near its surface is due to its original heat, it follows that at the present epoch at Paris this heat raises the temperature of the surface itself only by the fortieth part of a degree. Not knowing the radiating power of the substance of the globe, we cannot estimate the quantity of this initial heat which traverses in a given time from within to without an extent, also given, of the surface; but such would be its slowness in dissipating into space, that more than one thousand million of centuries must elapse to reduce the small increase of the fortieth of a degree to one half.
With regard to periodical inequalities, the relation which exists between each inequality at a given depth and the inequality corresponding to the exterior temperature is determined. Relations of this nature, for the knowledge of which we are indebted to M. Fourier, take place between the interior inequalities and those of the surface of the ground; these relations leave unknown the ratios of these latter inequalities to those of the outside which are the immediate data of the question.
The interior temperature to which the earth is subjected arises from three different sources, namely, from sidereal heat, from atmospherical heat, acting either by radiation or by contact, and from solar heat. These three sources of heat are successively examined. With regard to the first it is observed, that it is not at all probable that radiant heat emanating from the stars has the same intensity in all directions when it arrives at the earth. The experiments are indicated which it would be necessary to make in order to ascertain whether it really varies in the different regions of the sky. M. Melloni intends immediately to apply himself to these experiments, employing in them an extremely sensible instrument, of which he has made use in his researches on heat; a circumstance which cannot fail to lead to the solution of this important question of celestial physics.
Before considering the influence of atmospherical heat, I have formed a complete expression for the temperature, marked every instant by a thermometer suspended in the air, at any height above the surface of the earth exposed in the shade or in the direct rays of the sun. Although the greatest part of the quantities which this formula contains are unknown to us, many general consequences may however be deduced from it, which accord with experiment; it hence follows, that to determine the proper temperature of the air, it is necessary to employ the simultaneous observation of three thermometers, the surfaces of which are in a different state, and not two thermometers only, as is generally said. This formula also furnishes the means of comparing the temperatures indicated by different thermometers in relation to their radiating powers and to their property of absorbing the rays of the sun.
The mean of the annual temperatures, marked by a thermometer exposed in the open air and in the shade, forms the climateric temperature. It varies with the elevation of places above the level of the sea, and with the longitude and latitude, according to unknown laws. At Paris it is 10°·822, as M. Bouvard has concluded after 29 years of observations. There will be found in this Chapter a table of the mean temperatures for the twelve months of each of those years, which that gentleman has been pleased to communicate to us, and which had not before been published. It appears that in every point of the earth this climateric temperature differs very little from the mean temperature of the surface of the soil, as is shown by several examples. Notwithstanding, the variable temperature of this surface, and that which is marked at the same instant by a thermometer as little elevated above the surface as may be, are often very different from each other; it hence follows, that in a year the excess of the highest above the lowest temperature of the soil is at Paris nearly 24°, as will be seen in the course of this Chapter; and only about 17° for the thermometer suspended in the air and in the shade.
We now determine the part of exterior temperature which results from the atmospherical heat combined with sidereal heat. The necessary data for calculating its numerical value, à priori, being unknown to us, we show how this value, for every point of the globe, may be deduced from the mean temperature of its surface. At Paris this exterior temperature is 13°. Although we cannot determine separately the portion of this temperature of the earth which arises from the atmospherical heat, there is reason to think that it is also negative, so that the other portion arising from sidereal heat must be less than 13° below zero. If we suppose that radiant heat emanating from the stars falls in the same quantity on all points of the globe, this temperature, higher than 13°, will be that of space at the place where the earth is at this time. Without being able to assign the degree of heat of space, we may however admit, that its temperature differs little from zero, instead of being, as had been asserted, below the temperature of the coldest regions in the globe, and even of the freezing-point of mercury. As to the central temperature of the whole mass of the earth, even supposing its original heat to be entirely dissipated, and that it is no longer equal to the present temperature of space, we have no means of obtaining a knowledge of it.
According to a theorem of Lambert, the whole amount of solar heat which falls upon the earth is the same during different seasons, notwithstanding the inequality of their lengths, which is found to be compensated by that of the distances from the sun to the earth. This quantity of heat varies in the inverse ratio of the parameter of the ellipse described by the earth; it also varies with the obliquity of the ecliptic; but it does not appear that these variations can ever produce any considerable effect on the heat of the globe. The quantities of solar heat which fall in equal times upon the two hemispheres are nearly equal; but on account of the different states of their surfaces, those quantities are absorbed in different proportions; and the power of absorbing the rays of the sun increasing in a greater ratio than the radiating power, which is greater for dry land than for the sea, we conclude that the mean temperature of our hemisphere, where dry land is in a greater proportion, must be greater than that of the southern hemisphere; which agrees with observation.
The solar heat, which reaches each point of the globe, varies at different hours of the day; it is null when the sun is beneath the horizon; during the year it varies also with its declination; and the expression changes its form as the latitude of the point under consideration is greater or less than the complement of the obliquity of the ecliptic. I have therefore considered the part of the exterior temperature which arises from this source of heat as a discontinuous function of the horary angle, and of the longitude of the sun, to which I have applied the formulæ of the preceding Chapters, in order to convert it into series of sines and cosines of the multiples of these two angles. By this means I have obtained the complete expressions of the diurnal and annual inequalities of the temperature of the earth which arise from its double motion. These formulæ show, that at the equator the annual inequalities are much less than elsewhere; a circumstance which furnishes the explanation of a fact observed by M. Boussingault in his journey to the Cordilleras, and upon which he had relied in order to determine with great facility the climateric temperatures of the places which he visited. The same formulæ agree also, in a remarkable manner, with the temperatures which M. Arago has observed at Paris during many years, and at depths varying from two to eight metres (from 6·56 to 26·24 English feet).