The sum of the two actions of m on μ and of m on u' is therefore-2qcy.

Suppose then that the plane H belongs to the infinite solid whose temperature equation is v = A + cz, and that we denote also by m, μ and μ those molecules in this solid whose coordinates are x, y, z for the first, x+a, y + B, z+y for the second, and x-a, y - B, z+y for the third: we shall have, as in the preceding case, v — w+v — w' = — 2cy. Thus the sum of the two actions of m on μ and of m on μ', is the same in the infinite solid as in the prism enclosed between the six planes at right angles.

We should obtain a similar result, if we considered the action of another point n below the plane H on two others v and v', situated at the same height above the plane. Hence, the sum of all the actions of this kind, which are exerted across the plane H, that is to say the whole quantity of heat which, during unit of time, passes to the upper side of this surface, by virtue of the action of very near molecules which it separates, is always the same in both solids.

95. In the second of these two bodies, that which is bounded by two infinite planes, and whose temperature equation is v = A + cz, we know that the quantity of heat which flows during unit of time across unit of area taken on any horizontal section whatever is cK, c being the coefficient of z, and K the specific conducibility; hence, the quantity of heat which, in the prism enclosed between six planes at right angles, crosses during unit of time, unit of area taken on any horizontal section whatever, is also cK, when the linear equation which represents the temperatures of the prism is

v = A + ax + by + cz.

In the same way it may be proved that the quantity of heat which, during unit of time, flows uniformly across unit of area taken on any section whatever perpendicular to x, is expressed by - aK, and that the whole quantity which, during unit of time, crosses unit of area taken on a section perpendicular to y, is expressed by - bK.

The theorems which we have demonstrated in this and the two preceding articles, suppose the direct action of heat in the

interior of the mass to be limited to an extremely small distance, but they would still be true, if the rays of heat sent out by each molecule could penetrate directly to a quite appreciable distance, but it would be necessary in this case, as we have remarked in Article 70, to suppose that the cause which maintains the temperatures of the faces of the solid affects a part extending within the mass to a finite depth.

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SECTION VIII.

Measure of the movement of heat at a given point of a solid mass.

96. It still remains for us to determine one of the principal elements of the theory of heat, which consists in defining and in measuring exactly the quantity of heat which passes through every point of a solid mass across a plane whose direction is given.

If heat is unequally distributed amongst the molecules of the same body, the temperatures at any point will vary every instant. Denoting by t the time which has elapsed, and by the temperature attained after a time t by an infinitely small molecule whose co-ordinates are x, y, z; the variable state of the solid will be expressed by an equation similar to the following v = F(x, y, z, t). Suppose the function F to be given, and that consequently we can determine at every instant the temperature of any point whatever; imagine that through the point m we draw a horizontal plane parallel to that of x and y, and that on this plane we trace an infinitely small circle w, whose centre is at m; it is required to determine what is the quantity of heat which during the instant dt will pass across the circle w from the part of the solid which is below the plane into the part above it.

All points extremely near to the point m and under the plane exert their action during the infinitely small instant dt, on all those which are above the plane and extremely near to the point m, that is to say, each of the points situated on one side of this plane will send heat to each of those which are situated on the other side.

We shall consider as positive an action whose effect is to transport a certain quantity of heat above the plane, and as negative that which causes heat to pass below the plane. The

sum of all the partial actions which are exerted across the circle w, that is to say the sum of all the quantities of heat which, crossing any point whatever of this circle, pass from the part of the solid below the plane to the part above, compose the flow whose expression is to be found.

It is easy to imagine that this flow may not be the same throughout the whole extent of the solid, and that if at another point m' we traced a horizontal circle @ equal to the former, the two quantities of heat which rise above these planes w and w' during the same instant might not be equal: these quantities are comparable with each other and their ratios are numbers which may be easily determined.

97. We know already the value of the constant flow for the case of linear and uniform movement; thus in the solid enclosed between two infinite horizontal planes, one of which is maintained at the temperature a and the other at the temperature b, the flow of heat is the same for every part of the mass; we may regard it as taking place in the vertical direction only. The value correspond

ing to unit of surface and to unit of time is K (a = 5),

e

e denoting the perpendicular distance of the two planes, and K the specific conducibility: the temperatures at the different points of the

solid are expressed by the equation v = a - (a - b) z.

e

When the problem is that of a solid comprised between six rectangular planes, pairs of which are parallel, and the temperatures at the different points are expressed by the equation v = A + ax+by+cz,

the propagation takes place at the same time along the directions of x, of y, of z; the quantity of heat which flows across a definite portion of a plane parallel to that of x and y is the same throughout the whole extent of the prism; its value corresponding to unit of surface, and to unit of time is - cK, in the direction of z, it is -bK, in the direction of y, and aK in that of x.

In general the value of the vertical flow in the two cases which we have just cited, depends only on the coefficient of z and on

the specific conducibility K; this value is always equal to - K

dv dz

The expression of the quantity of heat which, during the instant dt, flows across a horizontal circle infinitely small, whose area is o, and passes in this manner from the part of the solid which is below the plane of the circle to the part above, is, for the two cases

98. It is easy now to generalise this result and to recognise that it exists in every case of the varied movement of heat expressed by the equation v = F(x, y, z, t).

Let us in fact denote by x', y', z', the co-ordinates of this point m, and its actual temperature by v. Let x' + §, y' + n, z' + 5, he the co-ordinates of a point μ infinitely near to the point m, and whose temperature is w; &, n, are quantities infinitely small added to the co-ordinates ', y', '; they determine the position of molecules infinitely near to the point m, with respect to three rectangular axes, whose origin is at m, parallel to the axes of x, y, and z. Differentiating the equation

v = f (x, y, z, t)

and replacing the differentials by g, n,, we shall have, to express the value of w which is equivalent to v+ dv, the linear equation

dv' dv' dv

w= v' + &+ η+ ; the coefficients v',

dx dy dz

dv dv dv

'dx' dy' dz, are functions of x, y, z, t, in which the given and constant values x', y', z′, which belong to the point m, have been substituted for x, y, z.

Suppose that the same point m belongs also to a solid enclosed between six rectangular planes, and that the actual temperatures of the points of this prism, whose dimensions are finite, are expressed by the linear equation w = A + a§ + bn + c; and that the molecules situated on the faces which bound the solid are maintained by some external cause at the temperature which is assigned to them by the linear equation. §, n, are the rectangular co-ordinates of a molecule of the prism, whose temperature is w, referred to three axes whose origin is at m.

This arranged, if we take as the values of the constant coefficients A, a, b, c, which enter into the equation for the prism, the dv' dv' dv which belong to the differential equadx' dy' dz'

quantities v',

tion; the state of the prism expressed by the equation

will coincide as nearly as possible with the state of the solid; that is to say, all the molecules infinitely near to the point m will have the same temperature, whether we consider them to be in the solid or in the prism. This coincidence of the solid and the prism is quite analogous to that of curved surfaces with the planes which touch them.

It is evident, from this, that the quantity of heat which flows in the solid across the circle w, during the instant dt, is the same as that which flows in the prism across the same circle; for all the molecules whose actions concur in one effect or the other, have the same temperature in the two solids. Hence, the flow in question, in one solid or the other, is expressed by K

dv

dz

wdt.

It would be K wdt, if the circle w, whose centre is m, were

dv dy

perpendicular to the axis of y, and – K

perpendicular to the axis of x.

dv da

wdt, if this circle were

The value of the flow which we have just determined varies in the solid from one point to another, and it varies also with the time. We might imagine it to have, at all the points of a unit of surface, the same value as at the point m, and to preserve this value during unit of time; the flow would then be expressed by - R it would be - K in the direction of y, and - Kdv

dv

dz'

dv

dy

dx

in that of x. We shall ordinarily employ in calculation this

value of the flow thus referred to unit of time and to unit of surface.

99. This theorem serves in general to measure the velocity with which heat tends to traverse a given point of a plane situated in any manner whatever in the interior of a solid whose temperatures vary with the time. Through the given point m, a perpendicular must be raised upon the plane, and at every point of this perpendicular ordinates must be drawn to represent the actual temperatures at its different points. A plane curve will thus be formed whose axis of abscissæ is the perpendicular.

F. H.

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