joins the extremities a and B; thus, denoting by z the height of an intermediate section or its perpendicular distance from the plane 4, by e the whole height or distance AB, and by v the temperature of the section whose height is 2, we must have the

In fact, if the temperatures were at first established in accordance with this law, and if the extreme surfaces A and B were always kept at the temperatures a and b, no change would happen in the state of the solid. To convince ourselves of this, it will be sufficient to compare the quantity of heat which would traverse an intermediate section A' with that which, during the same time, would traverse another section B'.

Bearing in mind that the final state of the solid is formed and continues, we see that the part of the mass which is below the plane A' must communicate heat to the part which is above that plane, since this second part is cooler than the first.

Imagine two points of the solid, m and m', very near to each other, and placed in any manner whatever, the one m below the plane A', and the other m' above this plane, to be exerting their action during an infinitely small instant: m the hottest point will communicate to m' a certain quantity of heat which will cross the plane A'. Let x, y, z be the rectangular coordinates. of the point m, and x, y, z' the coordinates of the point m': consider also two other points n and n' very near to each other, and situated with respect to the plane B', in the same manner in which m and m' are placed with respect to the plane A': that is to say, denoting by the perpendicular distance of the two sections A' and B, the coordinates of the point n will be x, y, z + S and those of the point n', x, y, z+; the two distances mm' and nn' will be equal: further, the difference of the temperature v of the point m above the temperature v' of the point m' will be the same as the difference of temperature of the two points n and n'. In fact the former difference will be determined by substituting first z and then z' in the general equation

and subtracting the second equation from the first, whence the

-(z-z). We shall then find, by the sub

e

stitution of +5 and 2+, that the excess of temperature of the point n over that of the point n' is also expressed by

It follows from this that the quantity of heat sent by the point m to the point m' will be the same as the quantity of heat sent by the point n to the point n', for all the elements which concur in determining this quantity of transmitted heat are the

same.

It is manifest that we can apply the same reasoning to every system of two molecules which communicate heat to each other across the section A' or the section B; whence, if we could sum up the whole quantity of heat which flows, during the same instant, across the section A' or the section B', we should find this quantity to be the same for both sections.

From this it follows that the part of the solid included between A' and B' receives always as much heat as it loses, and since this result is applicable to any portion whatever of the mass included between two parallel sections, it is evident that no part of the solid can acquire a temperature higher than that which it has at present. Thus, it has been rigorously demonstrated that the state of the prism will continue to exist just as it was at first.

Hence, the permanent temperatures of different sections of a solid enclosed between two parallel infinite planes, are represented by the ordinates of a straight line aß, and satisfy the linear b-a e

equation v = a + ·

2.

66. By what precedes we see distinctly what constitutes the propagation of heat in a solid enclosed between two parallel and infinite planes, each of which is maintained at a constant temperature. Heat penetrates the mass gradually across the lower plane: the temperatures of the intermediate sections are raised, but can never exceed nor even quite attain a certain limit which they approach nearer and nearer: this limit or final temperature is different for different intermediate layers, and

decreases in arithmetic progression from the fixed temperature of the lower plane to the fixed temperature of the upper plane.

The final temperatures are those which would have to be given to the solid in order that its state might be permanent; the variable state which precedes it may also be submitted to analysis, as we shall see presently: but we are now considering only the system of final and permanent temperatures. In the last state, during each division of time, across a section parallel to the base, or a definite portion of that section, a certain quantity of heat flows, which is constant if the divisions of time. are equal. This uniform flow is the same for all the intermediate sections; it is equal to that which proceeds from the source, and to that which is lost during the same time, at the upper surface of the solid, by virtue of the cause which keeps the temperature

constant.

67. The problem now is to measure that quantity of heat which is propagated uniformly within the solid, during a given time, across a definite part of a section parallel to the base: it depends, as we shall see, on the two extreme temperatures a and b, and on the distance e between the two sides of the solid; it would vary if any one of these elements began to change, the other remaining the same. Suppose a second solid to be formed of the same substance as the first, and enclosed between two

infinite parallel planes, whose perpendicular distance is e' (see fig. 2) the lower side is maintained at a fixed temperature a', and the upper side at the fixed temperature b'; both solids are considered to be in that final and permanent state which has the property of maintaining itself as soon as it has been formed.

F. H.

4

Thus the law of the temperatures is expressed for the first body

b- a

by the equation v=a+ z, and for the second, by the equa

tion u = a +

b' - a'

e

z, v in the first solid, and u in the second, being

the temperature of the section whose height is z.

This arranged, we will compare the quantity of heat which, during the unit of time traverses a unit of area taken on an intermediate section L of the first solid, with that which during the same time traverses an equal area taken on the section L' of the second, e being the height common to the two sections, that is to say, the distance of each of them from their own base. We shall consider two very near points n and n' in the first body, one of which n is below the plane L and the other n' above this plane: x, y, z are the co-ordinates of n : and x', y', z' the co-ordinates of n', e being less than z', and greater than z.

We shall consider also in the second solid the instantaneous action of two points p and p', which are situated, with respect to the section L', in the same manner as the points n and n' with respect to the section L of the first solid. Thus the same coordinates x, y, z, and x, y, z' referred to three rectangular axes in the second body, will fix also the position of the points p and p'.

Now, the distance from the point n to the point n' is equal to the distance from the point p to the point p', and since the two bodies are formed of the same substance, we conclude, according to the principle of the communication of heat, that the action of n on n', or the quantity of heat given by n to n', and the action of p on p', are to each other in the same ratio as the differences of the temperature v-v' and u-u'.

Substituting v and then v' in the equation which belongs to the first solid, and subtracting, we find v-v':

=

u

b-a

e

(z-z'); we

b' - a'

_

=

whence the ratio of the two actions in question is that of

b to

a' - b'

We may now imagine many other systems of two molecules, the first of which sends to the second across the plane L, a certain quantity of heat, and each of these systems, chosen in the first solid, may be compared with a homologous system situated in the second, and whose action is exerted across the section L'; we can then apply again the previous reasoning to prove that the α b a' - b' ratio of the two actions is always that of to é

e

Now, the whole quantity of heat which, during one instant, crosses the section L, results from the simultaneous action of a multitude of systems each of which is formed of two points; hence this quantity of heat and that which, in the second solid, crosses during the same instant the section L', are also to each a- b a'-b' other in the ratio of

e

to

It is easy then to compare with each other the intensities of the constant flows of heat which are propagated uniformly in the two solids, that is to say, the quantities of heat which, during unit of time, cross unit of surface of each of these bodies. The ratio of these intensities is that of the two quotients a' - b'

e

a-b

and

e

If the two quotients are equal, the flows are the same, whatever in other respects the values a, b, e, a', b', e', may be ; in general, denoting the first flow by F and the second by F', F α b a' - b'

we shall have

68. Suppose that in the second solid, the permanent temperature a of the lower plane is that of boiling water, 1; that the temperature e' of the upper plane is that of melting ice, 0; that the distance e' of the two planes is the unit of measure (a metre); let us denote by K the constant flow of heat which, during unit of time (a minute) would cross unit of surface in this last solid, if it were formed of a given substance; K expressing a certain number of units of heat, that is to say a certain number of times the heat necessary to convert a kilogramme of ice into water: we shall have, in general, to determine the