multitude of actions whose effects are added; but it is not from this cause that its value during unit of time is a finite and measurable magnitude, even although it be determined only by an extremely small difference between the temperatures.

When a heated body loses its heat in an elastic medium, or in a space free from air bounded by a solid envelope, the value of the outward flow is assuredly an integral; it again is due to the action of an infinity of material points, very near to the surface, and we have proved formerly that this concourse determines the law of the external radiation'. But the quantity of heat emitted during the unit of time would be infinitely small, if the difference of the temperatures had not a finite value.

In the interior of masses the conductive power is incomparably greater than that which is exerted at the surface. This property, whatever be the cause of it, is most distinctly perceived by us, since, when the prism has arrived at its constant state, the quantity of heat which crosses a section during the unit of time exactly balances that which is lost through the whole part of the heated surface, situated beyond that section, whose temperatures exceed that of the medium by a finite magnitude. When we take no account of this primary fact, and omit the divisor in the expression for the flow, it is quite impossible to form the differential equation, even for the simplest case; a fortiori, we should be stopped in the investigation of the general equations.

5th. Further, it is necessary to know what is the influence of the dimensions of the section of the prism on the values of the acquired temperatures. Even although the problem is only that of the linear movement, and all points of a section are regarded as having the same temperature, it does not follow that we can disregard the dimensions of the section, and extend to other prisms the consequences which belong to one prism only. The exact equation cannot be formed without expressing the relation. between the extent of the section and the effect produced at the extremity of the prism.

We shall not develope further the examination of the principles which have led us to the knowledge of the differential equations;

1 Mémoires de l'Académie des Sciences, Tome v. pp. 204-8. Communicated in 1811. [A. F.]

we need only add that to obtain a profound conviction of the usefulness of these principles it is necessary to consider also various difficult problems; for example, that which we are about to indicate, and whose solution is wanting to our theory, as we have long since remarked. This problem consists in forming the differential equations, which express the distribution of heat in fluids in motion, when all the molecules are displaced by any forces, combined with the changes of temperature. The equations which we gave in the course of the year 1820 belong to general hydrodynamics; they complete this branch of analytical mechanics'.

430. Different bodies enjoy very unequally the property which physicists have called conductibility or conducibility, that is to say, the faculty of admitting heat, or of propagating it in the interior of their masses. We have not changed these names, though they

1 See Mémoires de l'Académie des Sciences, Tome XII. Paris, 1833, pp. 515–530. In addition to the three ordinary equations of motion of an incompressible fluid, and the equation of continuity referred to rectangular axes in direction of which the velocities of a molecule passing the point x, y, z at time t are u, v, w, its temperature being 0, Fourier has obtained the equation

in which K is the conductivity and C the specific heat per unit volume, as follows.

Into the parallelopiped whose opposite corners are (x, y, z), (x + Ax, y + Ay, z+ Az), the quantity of heat which would flow by conduction across the lower face Ax▲y, if the fluid were at rest, would be - K Ar Ay At in time At, and the gain by

do
dz

convection + Cw Ar Ay At; there is a corresponding loss at the upper face Ax Ay;

hence the whole gain is, negatively, the variation of (– K

respect to z, that is to say, the gain is equal to

do

dz

+Cwo) Ax Ay At with

Two similar expressions denote the gains in direction of y and z; the sum of the

do dt

three is equal to C At Ax Ay Az, which is the gain in the volume Az Ay Az

in time At: whence the above equation.

The coefficients K and C vary with the temperature and pressure but are usually treated as constant. The density, even for fluids denominated incompressible, is subject to a small temperature variation.

It may be noticed that when the velocities u, v, w are nul, the equation reduces to the equation for flow of heat in a solid.

It may also be remarked that when K is so small as to be negligible, the equation has the same form as the equation of continuity. [A. F.]

do not appear to us to be exact. Each of them, the first especially, would rather express, according to all analogy, the faculty of being conducted than that of conducting.

Heat penetrates the surface of different substances with more or less facility, whether it be to enter or to escape, and bodies are unequally permeable to this element, that is to say, it is propagated in them with more or less facility, in passing from one interior molecule to another. We think these two distinct properties might be denoted by the names penetrability and permeability'.

Above all it must not be lost sight of that the penetrability of a surface depends upon two different qualities: one relative to the external medium, which expresses the facility of communication by contact; the other consists in the property of emitting or admitting radiant heat. With regard to the specific permeability, it is proper to each substance and independent of the state of the surface. For the rest, precise definitions are the true foundation of theory, but names have not, in the matter of our subject, the same degree of importance.

431. The last remark cannot be applied to notations, which contribute very much to the progress of the science of the Calculus. These ought only to be proposed with reserve, and not admitted but after long examination. That which we have employed reduces itself to indicating the limits of the integral above and below the sign of integration ; writing immediately after this sign the differential of the quantity which varies between these limits.

We have availed ourselves also of the sign Σ to express the sum of an indefinite number of terms derived from one general term in which the index i is made to vary. We attach this index if necessary to the sign, and write the first value of i below, and the last above. Habitual use of this notation convinces us of

1 The coefficients of penetrability and permeability, or of exterior and interior conduction (h, K), were determined in the first instance by Fourier, for the case of cast iron, by experiments on the permanent temperatures of a ring and on the h varying temperatures of a sphere. The value of by the method of Art. 110, K

and the value of h by that of Art. 297. Mem. de l'Acad. d. Sc. Tome v. pp. 165, 220, 228. [A. F.]

the usefulness of it, especially when the analysis consists of definite integrals, and the limits of the integrals are themselves the object of investigation.

432. The chief results of our theory are the differential equations of the movement of heat in solid or liquid bodies, and the general equation which relates to the surface. The truth of these equations is not founded on any physical explanation of the effects of heat. In whatever manner we please to imagine the nature of this element, whether we regard it as a distinct material thing which passes from one part of space to another, or whether we make heat consist simply in the transfer of motion, we shall always arrive at the same equations, since the hypothesis which we form must represent the general and simple facts from which the mathematical laws are derived.

The quantity of heat transmitted by two molecules whose temperatures are unequal, depends on the difference of these temperatures. If the difference is infinitely small it is certain that the heat communicated is proportional to that difference; all experiment concurs in rigorously proving this proposition. Now in order to establish the differential equations in question, we consider only the reciprocal action of molecules infinitely near. There is therefore no uncertainty about the form of the equations. which relate to the interior of the mass.

The equation relative to the surface expresses, as we have said, that the flow of the heat, in the direction of the normal at the boundary of the solid, must have the same value, whether we calculate the mutual action of the molecules of the solid, or whether we consider the action which the medium exerts upon the envelope. The analytical expression of the former value is very simple and is exactly known; as to the latter value, it is sensibly proportional to the temperature of the surface, when the excess of this temperature over that of the medium is a sufficiently small quantity. In other cases the second value must be regarded as given by a series of observations; it depends on the surface, on the pressure and on the nature of the medium; this observed value ought to form the second member of the equation relative to the surface.

In several important problems, the equation last named is re

placed by a given condition, which expresses the state of the surface, whether constant, variable or periodic.

433. The differential equations of the movement of heat are mathematical consequences analogous to the general equations of equilibrium and of motion, and are derived like them from the most constant natural facts.

The coefficients c, h, k, which enter into these equations, must be considered, in general, as variable magnitudes, which depend on the temperature or on the state of the body. But in the application to the natural problems which interest us most, we may assign to these coefficients values sensibly constant.

The first coefficient c varies very slowly, according as the temperature rises. These changes are almost insensible in an interval of about thirty degrees. A series of valuable observations, due to Professors Dulong and Petit, indicates that the value of the specific capacity increases very slowly with the temperature.

The coefficient h which measures the penetrability of the surface is most variable, and relates to a very composite state. It expresses the quantity of heat communicated to the medium, whether by radiation, or by contact. The rigorous calculation of this quantity would depend therefore on the problem of the movement of heat in liquid or aeriform media. But when the excess of temperature is a sufficiently small quantity, the observations prove that the value of the coefficient may be regarded as constant. In other cases, it is easy to derive from known experiments a correction which makes the result sufficiently exact.

It cannot be doubted that the coefficient k, the measure of the permeability, is subject to sensible variations; but on this important subject no series of experiments has yet been made suitable for informing us how the facility of conduction of heat changes with the temperature' and with the pressure. We see, from the observations, that this quality may be regarded as constant throughout a very great part of the thermometric scale. But the same observations would lead us to believe that the value of the coefficient in question, is very much more changed by increments of temperature than the value of the specific capacity.

Lastly, the dilatability of solids, or their tendency to increase

1 Reference is given to Forbes' experiments in the note, p. 84. [A. F.]

F. H.

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