PlanetPhysics/Preliminary Definitions and General Notions

Preliminary definitions and general notions.

From The Analytical Theory of Heat by Joseph Fourier

{\mathbf 22.} Of the nature of heat uncertain hypotheses only could be formed, but the knowledge of the mathematical laws to which its effects are subject is independent of all hypothesis; it requires only an attentive examination of the chief facts which common observations have indicated, and which have been confirmed by exact experiments.

It is necessary then to set forth, in the first place, the general results of observation, to give exact definitions of all the elements of the analysis, and to establish the principles upon which this analysis ought to be founded.

The action of heat tends to exapand all bodies, solid, liquid or gaseous; this is the property which gives evidence of its presence. Solids and liquids increase in volume, if the quantity of heat which they contain increases; they contract if it diminishes.

When all the parts of a solid homogeneous body, for example those of a mass of metal, are equally heated, and presrve without any change the same quantity of heat, they have also and retain the same density. This state is expressed by saying that throughout the whole extent of the mass the molecules have a common and permanent temperature.

{\mathbf 23.} The termometer is a body whose smallest changes of volume can be appreciated; it serves to measure temperatures by the dilatation of a fluid or of air. We assume the construction, and use and properties of this instrument to be accurately known. The temperature of a body equally heated in every part, and which keeps its heat, is that which the thermometer indicates when it is and remains in perfect contact with the body in question.

Perfect contact is when the thermometer is completely immersed in a fluid mass, and, in general, when there is no point of the external surface of the instrument which is not touched by one of the points of the solid or liquid mass whose temperature is to be measured. In experiments it is not always necessary that this condition should be rigorously observed; but it ought to be assumed in order to make the definition exact.

{\mathbf 24.} Two fixed temperatures are determined on, namely: the temperature of melting ice which is denoted by 0, and the temperature of boiling water which we will denote by 1: the water is supposed to be boiling under an atmospheric pressure represented by a certain height of the barometer (76 centimetres), the mercury of the barometer being at the temperature 0.

{\mathbf 25.} Different quantities of heat are measured by determining how many times they contain a fixed quantity which is taken as the unit. Suppose a mass of ice having a definite weight (a kilogramme) to be at temperature 0, and to be converted into water at the same termperature 0 by the addition of a certain quantity of heat: the quantity of heat thus added is taken as the unit of measure. Hence the quantity of heat expressed by a number $C$ contains $C$ times the quantity required to dissolve a kilogramme of ice at the temperature zero into a mass of water at the same zero temperature.

{\mathbf 26.} To raise a metallic mass having a certain weight, a kilogramme of iron for example, from the temperature 0 to the temperature 1, a new quantity of heat must be added to that which is already contained in the mass. The number $C$ which denotes this additional quantity of heat, is the specific capacity of iron for heat; the number $C$ has very different values for different substances.

{\mathbf 27.} If a body of definite nature and weight (a kilogramme of mercury) occupies a volume $V$ at temperature 0, it will occupy a greater volume $V+\Delta$, when it has acquired the temperature 1, that is to say, when the heat which it contained at the temperature 0 has been increased by a new quantity $C$, equal to be specific capacity of the body for heat. But if, instead of adding this quantity $C$, a quantity $zC$ is added ($z$ being a number positive or negative) the new volume will be $V+\delta$ instead of $V+\Delta$. Now experiments shew that if $z$ is equal to ${\frac {1}{2}}$, the increase of volume $\delta$ is only half the total increment $\Delta$, and that in general the value of $\delta$ is $z\Delta$, when the quantity of heat added is $zC$.

{\mathbf 28.} The ratio $z$ of the two quantities $zC$ and $C$ of heat added, which is the same as the ratio of the two increments of volume $\delta$ and $\Delta$, is that which is called the temperature ; hence the quantity which expresses the actual temperature of a body represents the excess of its actual volume over the volume which it would occupy at the temperature of melting ice, unity representing the whole excess of volume which corresponds to the boiling point of water, over the volume which corresponds to the melting point of ice.

{\mathbf 29.} The increments of volume of bodies are in general proportional to the increments of the quantities of heat which produce the dilatations, but it must be remarked that this proportion is exact only in the case where the bodies in question are subjected to temperatures remote from those which determine their change of state. The application of these results to all liquids must not be relied on; and with respect to water in particular, dilatations do not always follow augmentations of heat.

In general the temperatures are numbers proportional to the quantities of heat added, and in the cases considered by us, these numbers are proportional also to the increments of volume.

{\mathbf 30.} Suppose that a body bounded by a plane surface having certain area (a square metre) is maintained in any manner whatever at constant temperature 1, common to all its points, and that the surface in question is in contact with air maintained at temperature 0: the heat which escapes continuously at the surface and passes into the surrounding medium will be replaced always by the heat which proceeds from the constant cause to whose action the body is exposed; thus, a certain quantity of heat denoted by $h$ will flow through the surface in a definite time (a minute).

This amount $h$, of a flow continuous and always similar to itself, which takes place at a unit of surface at a fixed temperature, is the measure of the external conducibility of the body, that is to say, of the facility with which its surface transmits heat to the atmospheric air.

The air is supposed to be continually displaced with a given uniform velocity: but if the velocity of the current increased, the quantity of heat communicated to the medium would vary also: the same would happen if the density of the medium were increased.

{\mathbf 31.} If the excess of the constant temperature of the body over the temperature of surrounding bodies, instead of being equal to 1, as has been supposed, had a less value, the quantity of heat dissipated would be less than $h$. The result of observation is, as we shall see presently, that this quantity of heat lost may be regarded as sensibly proportional to the excess of the temperature of the body over that of the air and surrounding bodies. Hence the quantity $h$ having been determined by one experiment in which the surface heated is at temperature 1, and the medium at temperature 0; we conclude that $hz$ would be the quantity, if the temperature of the surface were $z$ all the other circumstances remaining the same. This result must be admitted when $z$ is a small fraction.

{\mathbf 32.} The value $h$ of the quantity of heat which is dispersed across a heated surface is different for different bodies; and it varies for the same body according to the different states of the surface. The effect of irradiation diminishes as the surface becomes more polished; so that by destroying the polish of the surface the value of $h$ is considerably increased. A heated metallic body will be more quickly cooled if its external surface is covered with a black coating such as will entirely tarnish its metallic lustre.

{\mathbf 33.} The rays of heat which escape from the surface of a body pass freely through spaces void of air; they are propagated also in atmospheric air: their directions are not disturbed by agitations in the intervening air: they can be reflected by metal mirrors and collected at their foci. Bodies at a high temperature, when plunged into a liquid, heat directly only those parts of the mass with which their surface is in contact. The molecules whose distance from this surface is not extremely small, receive no direct heat; it is not the same with aeriform fluids; in these the rays of heat are borne with extreme rapidity to considerable distances, whether it be that part of these rays traverses freely the layers of air, or whether these layers transmit the rays suddenly without altering their direction.

{\mathbf 34.} When the heated body is placed in air which is maintained at a sensibly constant temperature, the heat communicated to the air makes the layer of the fluid nearest to the surface of the body lighter; this layer rises more quicly the more intensely it is heated, and is replaced by another mass of cool air. A current is thus established in the air whose direction is vertical, and whose velocity is greater as the temperature of the body is higher. For this reason of the body cooled itself gradually the velocity of the current would diminish with the termperature, and the law of cooling would not be exactly the same as if the body were exposed to a current of air at a constant velocity.

{\mathbf 35.} When bodies are sufficiently heated to diffuse a vivid light, part of their radiant heat mixed with that light can traverse transparent solids or liquids, and is subject to the force which produces refraction. The quantity of heat which possesses this faculty becomes less as the bodies are less inflamed; it is, we may say, insensible for very opaque bodies however highly they may be heated. A thin transparent plate intercepts almost all the direct heat which proceeds from an ardent mass of metal; but it becomes heated in proportion as the intercepted rays are accumulated in it; whence, if it is formed of ice, it becomes liquidl but if this plate of ice is exposed to the rays of a torch it allows a sensible amount of heat to pass through with the light.

{\mathbf 36.} We have taken as the measure of the external conducibility of a solid body a coefficient $h$, which denotes the quantity of heat which would pass, in a definite time (a minute), from the surface of this body, into atmospheric air, supposing that the surface had a definite extent (a square metre), that the constant temperature of the body was 1, and that of the air 0, and that the heated surface was exposed to a current of sir of a given invariable velocity. This value of $h$ is determined by observation. The quantity of heat expressed by the coefficient is composed of two distinct parts which cannot be measured except by very exact experiments. One is the heat communicated by way of contact to the surrounding air: the other, much less than the first, is the radiant heat emitted. We must assume, in our first investigations, that the quantity of heat lost does not change when the temperatures of the body and of the medium are augmented by the same sufficiently small quantity.

{\mathbf 37.} Solid substances differ again, as we have already remarked, by their property of being more or less permeable to heat; this quality is their conducibility proper: we shall give its definition and exact measure, after having treated of the uniform and linear propagation of heat. Liquid substances possess also the property of transmitting heat from molecule to molecule, and the numerical value of their conducibility varies according to the nature of the substances: but this effect is observed with difficulty in liquids, since their molecules change places on change of temperature. The propagation of heat in them depends chiefly on this continual displacement, in all cases where the lower parts of the mass are most exposed to the action of the source of heat. If, on the contrary, the source of heat be applied to that part of the mass which is hightes, as was the case in several of our experiments, the transfer of heat, which is very slow, does not produce any displacement, at least when the increase of temperature does not diminish the volume, as is indeed noticed in singular cases bordering on changes of state.

{\mathbf 38.} To this explanation of the chief results of observation, a general remark must be added on equilibrium of temperatures; which consists in this, that different bodies placed in the same region, all of whose parts are and remain equally heated, acquire also a common and permanent temperature.

Suppose that all the parts of a mass $M$ have a common and constant temperature $a$, which is maintained by any cause whatever: if a smaller body $m$ be placed in perfect contact with the mass $M$, it will assume the common temperature $a$.

In reality this result would not strictly occur except after an infinite time: but the exact meaning of the proposition is that if the body $m$ had the temperature $a$ before being placed in contact, it would keep it without any change. The same would be the case with a multitude of other bodies $n,p,q,r$ each of which was placed separately in perfect contact with the mass $M$: all would acquire the constant temperature $a$. Thus a thermometer if sucessively applied to the different bodies $m,n,p,q,r$ would indicate the same temperature.

{\mathbf 39.} The effect in question is independent of contact, and would still occur, if every part of the body $m$ were enclosed in the solid $M$, as in an enclosure, without touching any of its parts. For example, if the solid were a spherical envelope of a certain thickness, maintained by some external cause at a temperature $a$, and containing a space entirely deprived of air, and if the body $m$ could be placed in any part whatever of this spherical space, without touching any point of the internal surface of the enclosure, it would acquire the common temperature $a$, or rather, it would preserve it if it had it already. The result would be the same for all the other bodies $n,p,q,r,$ whether they were placed separately or all together in the same enclosure, and whatever also their substance and form might be.

{\mathbf 40.} Of all modes of presenting to ourselves the action of heat, that which seems simplest and most conformable to observation, consists in comparing this action to that of light. Molecules separated from one another reciprocally communicate, across empty space, their rays of heat, just as shining bodies transmit their light.

If within an enclosure closed in all directions, and maintained by some external cause at a fixed temperature $a$, we suppose different bodies to be placed without touching any part of the boundary, different effects will be ovserved according as the bodies, introduced into this space free from air, are more or less heated. If, in the first instance, we insert only one of these bodies, at the same temperature as the enclosure, it will send from all points of its surface as much heat as it receives from the solid which surrounds it, and is maintained in its orginal state by this exchange of equal quantities.

If we insert a second body whose temperature $b$ is less than $a$, it will at first receive from the surfaces which surround it on all sides without touching it, a quantity of heat greater than that which it gives out: it will be heated more and more and will absorb through its surface mroe heat than in the first instance.

The initial temperature $b$ continually rising, will approach without ceasing the fixed temperature $a$, so that after a certain time the difference will be almost insensible. The effect would be opposite if we placed within the same enclosure a third body whose temperature was greater than $a$.

{\mathbf 41.} All bodies have the property of emitting heat through their surface; the hotter they are the more they emit; the intensity of the emitted rays changes very considerably with the state of the surface.

{\mathbf 42.} Every surface which receives rays of heat from surrounding bodies reflects part and admits the rest: the heat which is not reflected, but introduced through the surface, accumulates within the solid; and so long as it exceeds the quantity dissipated by irradiation, the temperature rises.

{\mathbf 43.} The rays which tend to go out of heated bodies are arrested at the surface by a force which reflects part of them into the interior of the mass. The cause which hinders the incident rays from traversing the surface, and which divides these rays into two parts, of which one is reflected and the other admitted, acts in the same manner on the rays which are directed from the interior of the body towards external space.

If by bodifying the state of the surface we increase the force by which it reflects the incident rays, we increase at the same time the power which it has of reflecting towards the interior of the body rays which are tending to go out. The incident rays introduced into the mass, and the rays emitted through the surface, are equally diminished in quantity.

{\mathbf 44.} If within the enclosure above mentioned a number of bodies were placed at the same time, separate from each other and unequally heated, they would receive and transmit rays of heat so that at each exchange their temperatures would continually vary, and would all tend t obe come equal to the fixed temperature of the enclosure.

This effect is precisely the same as that which occurs when heat is propagated within solid bodies; for the molecules which compose these bodies are separated by spaces void of air, and have the property of receiving, accumulating and emitting heat. Each of them sends out rays on all sideds, and at the same time receives other rays from the molecules which surround it.

{\mathbf 45.} The heat given out by a point situated in the interior of a solid mass can pass directly to an extremely small distance only; it is, we may say, intercepted by the nearest particles; these particles only receive the heat directly and act on more distant points. It is different with gaseous fluids; the direct effects of radiation become sensible in them at very considerable distances.

{\mathbf 46.} Thus the heat which escapes in all directions from a part of the surface of a solid, passes on in air to very distant points; but is emitted only by those molecules of the body which are extremely near the surface. A point of a heated mass situated at a very small distance from the plane superficies which separates the mass from external space, sends to that space an infinity of rays, but they do not all arrive there; they are diminished by all that quantitiy of heat which is arrested by the intermediate molecules of the solid. The part of the ray actually dispersed into space becomes less according as it traverses a longer path within the mass. Thus the ray which escapes perpendicular to the surface has greater intensity then that which, departing from the same point, follows an obliquie direction, and the most oblique rays are wholly intercepted.

The same consequences apply to all the points which are near enough to the surface to take part in the emission of heat, from which it necessarily follows that the whole quantity of heat which escapes from the surface in the normal direction is very much greater than that whose direction is oblique. We have submitted this question to calculation, and our analysis proves that the intensity of the ray is proportional to the sine of the angle which the ray makes with the element of surface. Experiments had already indicated a similar result.

{\mathbf 47.} This theorem expresses a general law which has a necessary connection with the equilibrium and mode of action of heat. If the rays which escape from a heated surface had the same intensity in all directions, a thermometer placed at one of the points of a space bounded on all sides by an enclosure maintained at a constant temperature would indicate a temperature incomparably greater than that of the enclosure[1]. Bodies placed within this enclosure would not take a common temperature, as is always noticed; the temperature acquired by them would depend on the place which they occupied, or on their form, or on the forms of neighboring bodies.

The same results would be observed, or other effects equally opposed to common exerience, if beeween the rays which escape from the same point any other relations were admitted different from those which we have enunciated. We have recognised this law as the only one compatible with general fact of the equilibrium of radiant heat.

{\mathbf 48.} If a space free from air is bounded on all sides by a solid enclosure whose parts are maintained at a common and constant temperature $a$, and if a termometer, having the actual temperature $a$, is placed at any point whatever of the space, its temperature will continue without any change. It will receive therefore at each instant from the inner surface of the enclosure as much heat as it gives out to it. This effect of the rays of heat in a given space is, properly speaking, the measure of the temperature: but this consideration presupposes the mathematical theory of radiant heat.

If now between the thermometer and a part of the surface of the enclosure a body $M$ be placed whose temperature is $a$, the thermometer will cease to receive rays from one part of the inner surface, but the rays will be replaced by those which it will receive from the interposed body $M$. An easy calculation proves that the compensation is exact, so that the state of the thermometer will be unchanged. It is not the same if the temperature of the body $M$ is different from that of the enclosure. When it is greater, the rays which the interposed body $M$ sends to the thermometer and which replace the intercepted rays convey more heat than the latter; the temperature of the thermometer must therefore rise.

If, on the contrary, the intervening body has a temperature less than $a$, that of the thermometer must fall; for the rays which this body intercepts are replaced by those which it gives out, that is to say, by rays cooler than those of the enclosure; thus the thermomter does not receive all the heat necessary to maintain its temperature $a$.

{\mathbf 49.} Up to this point abstraction has been made of the power which all surfaces have of reflecting part of the rays which are sent to them. If this property were disregarded we should have only a very incomplete idea of the equilibrium of radiant heat.

Suppose then that on the inner surface of the enclosure, maintained at a constant temperature, there is a portion which enjoys, in a certain degree, the power in question; each of the reflecting surface will send into space two kinds of rays; the one go out from the very interior of the substance of which the enclosure is formed, the others are merely reflected by the same surface against which they had been sent. But at the same time that the surface repels on the outside part of the incident rays, it retains in the inside part of its own rays. In this respect an exact compensation is established, that is to say, every one of its own rays which the surface hinders from going out is replaced by a reflected ray of equal intensity.

The same result would happen, if the power of reflecting rays affected in any degree whatever other parts of the enclosure, or the surface of bodies placed within the same space and already at the common temperature.

Thus the reflection of heat does not disturb the equilibrium of temperatures, and does not introduce, whilst that equilibrium exists, any change in the law according to which the intensity of rays which leave the same point decreases proportionally to the sine of the angle of emission.

{\mathbf 50.} Suppose that in the same enclosure, all of whose parts maintain the temperature $a$, we place an isolated body $M$, and a polished metal surface $R$, which, turning its concavity towards the body, reflects great part of the rays which it received from the body; if we place a thermometer between the body $M$ and the reflecting surface $R$, at the focus of this mirror, three different effects will be oserved according as the temperature of the body $M$ is equal to the common temperature $a$, or is greater or less.

In the first case, the thermometer preserves the temperature $a$; it receives $1^{0}$, rays of heat from all parts of the enclosure not hidden from it by the body $M$ or by the mirror; $2^{0}$, rays given out by the body; $3^{0}$, those which the surface $R$ sends out to the focus, whether they come from the mass of the mirror itself, or whether its surface has simply reflected them; and amongst the last we may distinguish between those which have been sent to the mirror by the mass $M$, and those which it has received from the enclosure. All the rays in question proceed from surfaces which, by hypothesis, have a common temperature $a$, so that the thermometer is precisely in the same state as if the space bounded by the enclosure contained no other body but itself.

In the second case, the thermometer placed between the heated body $M$ and the mirror, must acquire a temperature greater than $a$. In reality, it receives the same rays as in the first hypothesis; but with two remarkable differences: one arises from the fact that the rays sent by the body $M$ to the mirror, and reflected upon the thermometer, contain more heat than in the first case. The other difference depends on the fact that the rays sent directly by the body $M$ to the thermometer contain more heat than formerly. Both causes, and chiefly the first, assist in raising the temperature of the thermometer.

In the third case, that is to say, when the temperature of the mass $M$ is less than $a$, the temperature must assume also a temperature less than $a$. In fact, it receives again all the varieties of rays which we distinguished in the first case: but there are two kinds of them which contain less heat than in this first hypothesis, that is to say, those which, being sent out by the body $M$, are reflected by the mirror upon the thermometer, and those which the same body $M$ sends to it directly. Thus the thermometer does not receive all the heat which it requires to preserve its orginal termperature $a$. It gives out more heat than it receives. It is inevitable then that its temperature must fall to the point at which the reays which it receives suffice to compensate those which it loses. This last effect is what is called the reflection of cold, and which, properly speaking, consists in the reflection of too feeble heat. The mirror intercepts a certain quantity of heat, and replaces it by a less quantity.

{\mathbf 51.} If in the enclosure, maintained at a constant temperature $a$, a body $M$ be placed, whose temperature $a'$ is less than $a$, the presence of this body will lower the thermometer exposed to its rays, and we may remark that the rays sent to the thermometer from the surface of the body $M$, are in general of two kinds, namely, those which come from inside the mass $M$, and those which, coming from different parts of the enclosure, meet the surface $M$ and are reflected upon the thermometer. The latter rays have the common temperature $a$, but those which belong to the body $M$ contain less heat, and these are the rays which cool the thermometer. If now, by changing the state of the surface of the body $M$, for example, by destroying the polish, we diminish the power which it has of reflecting the incident rays, the thermometer will fall still lower, and will assume a temperature $a''$ less than $a$. In fact all the conditions would be the same as in the preceding case, if ti were not that the body $M$ gives out a greater quantity of its own rays and reflects a less quantity of the rays which it receives from the enclosure; that is to say, these last rays, which have the common temperature, are in part replaced by cooler rays. Hence the thermometer no longer receives so much heat as formerly.

If, independently of the change in the surface of the body $M$, we place a metal mirror adapted to reflect upon the thermometer the rays which have left $M$, the temperature will assume a value $a'''$ less than $a''$. The mirror, in fact, intercepts from the thermometer part of the rays of the enclosure which all have the temperature $a$, and replaces them by three kinds of rays; namely, $1^{0}$, those which come from the interior of the mirror itself, and which have the common temperature; $2^{0}$, those which the different parts of the enclosure send to the mirror with the same temperature, and which are reflected to the focus; $3^{0}$, those which, coming from the interior of the body $M$, fall upon the mirror, and are reflected upon the thermometer. The last rays have a temperature less than $a$; hence the thermometer no longer receives so much heat as it received before the mirror was set up.

Lastly, if we proceed to change also the state of the surface of the mirror, and by giving it a more perfect polish, increase its power of reflecting heat, the thermometer will fall still lower,. In fact, all the conditions exist which occurred in the preceding case. Only, it happens that the mirror gives out a less quantity of its own rays, and replaces them by those which it reflets. Niw, amongst these last rays, all those which proceed from the interior of the mass $M$ are less intense than if they had come from the interior of the metal mirror; hence the thermometer receives still less heat than formerly: it will assume therefore a temperature $a''''$ less than $a'''$.

By the same principles all the known facts of the radiation of heat or of cold are easily explained.

{\mathbf 52.} The effects of heat can by no means be compared with those of an elastic fluid whose molecules are at rest.

It would be useless to attempt to deduce from this hypothesis the laws of propagation which we have explained in this work, and which all experience has confirmed. The free state of heat is the same as that of light; the active state of this element is then entirely different from that of gaseous substances. Heat acts in the same manner in a vacuum, in elastic fluids, and in liquid or solid masses, it is propagated only by way of radiation, but its sensible effects differ according to the nature of bodies.

{\mathbf 53.} Heat is the origin of all elasticity; it is the repulsive force which preserves the form of solid masses, and the volume of liquids. In solid masses, neighbouring molecules would yield to their mutual attraction, if its effect were not destroyed by the heat which separates them.

This elastic force is greater according as the temperature is higher; which is the reason why bodies dilate or contract when their termperature is raised or lowered.

{\mathbf 54.} The equilibrium which exists, in the interior of a solid mass, between the repulsive force of heat and the molecular attraction, is stable; that is to say, it re-establishes itself when disturbed by an accidental cause. If the moleculres are arranged at distances proper for equilibrium, and if an external force begins to increase this distance without any change of temperature, the effect of attraction begins by surpassing that of heat, and brings back the molecules to their original position, after a multitude of oscillations which become less and less sensible.

A similar effect is exerted in the opposite sense when a mechanical cause diminishes the primitive distance of the molecules; such is the origin of the vibrations of sonorous or flexible bodies, and of all the effects of their elasticity.

{\mathbf 55.} In the liquid or gaseous state of matter, the external pressure is additional or supplementary to the molecular attractin, and, acting on the surface, does not oppose change of form but only change of the volume occupied. Analytical investigation will best shew how the repulsive force of heat, opposed to the attraction of the molecules or to the external pressure, assists in the composition of bodies, solid or liquid, formed of one or more elements, and determines the elastic properties of gaseous fluids; but these researches do not belong to the object before us, and appear in dynamic theories.

{\mathbf 56.} It cannot be doubted that the mode of action of heat always consists, like that of light, in the reciprocal communication of rays, and this explanation is at the present time adopted by the majority of physicists; but it is not necessary to consider the phenomena under this aspect in order to establish the theory of heat. In the course of this work it will be seen how the laws of equilibrium and propagation of radiant heat, in solid or liquid masses, can be rigorously demonstrated, independently of any physcial explanation, as the necessary consequences of common observations.

References

This chapter is a derivative from the public domain work in [2].

[1] See proof by M. Fourier. Ann. d. Ch. et Ph. Ser. 2, IV. p. 128.

[2] Fourier, Joseph. "The Analytcal Theory of Heat" Translation by Alexander Freeman, Cambridge: At the University Press, London, 1878.