fixed point o, and by the ordinate which represents the temperature of the point p, v will vary with the distance and will be a certain function f(e) of that distance; the quantity of heat which would flow across the circle w, placed at the dv

point p perpendicular to the line, will be - K wdt, or

de

We may express this result in the following manner, which facilitates its application.

To obtain the actual flow of heat at a point p of a straight line drawn in a solid, whose temperatures vary by action of the molecules, we must divide the difference of the temperatures at two points infinitely near to the point p by the distance between these points. The flow is proportional to the quotient.

142. THEOREM IV. From the preceding Theorems it is easy to deduce the general equations of the propagation of heat.

Suppose the different points of a homogeneous solid of any form whatever, to have received initial temperatures which vary successively by the effect of the mutual action of the molecules, and suppose the equation v = f (x, y, z, t) to represent the successive states of the solid, it may now be shewn that v a function of four variables necessarily satisfies the equation

In fact, let us consider the movement of heat in a molecule enclosed between six planes at right angles to the axes of x, y, and z; the first three of these planes pass through the point m whose coordinates are x, y, z, the other three pass through the point m', whose coordinates are x + dx, y + dy, z + dz.

dv

During the instant dt, the molecule receives, across the lower rectangle dady, which passes through the point m, a quantity of heat equal to - K dx dy dz dt. To obtain the quantity which escapes from the molecule by the opposite face, it is sufficient to change z into z+dz in the preceding expression,

that is to say, to add to this expression its own differential taken with respect to z only; we then have

as the value of the quantity which escapes across the upper rectangle. The same molecule receives also across the first rectangle dz dx which passes through the point m, a quantity

dv

of heat equal to -Kydz dx dt; and if we add to this ex

pression its own differential taken with respect to y only, we find that the quantity which escapes across the opposite face dz dx is expressed by

Lastly, the molecule receives through the first rectangle dy dz

dv

a quantity of heat equal to K ddy dz dt, and that which it loses across the opposite rectangle which passes through m' is expressed by

We must now take the sum of the quantities of heat which the molecule receives and subtract from it the sum of those which it loses. Hence it appears that during the instant dt, a total quantity of heat equal to

accumulates in the interior of the molecule.

It remains only to obtain the increase of temperature which must result from this addition of heat.

D being the density of the solid, or the weight of unit of volume, and C the specific capacity, or the quantity of heat which raises the unit of weight from the temperature 0 to the temperature 1; the product CDdx dy dz expresses the quantity

F. H.

8.

of heat required to raise from 0 to 1 the molecule whose volume is dx dy dz. Hence dividing by this product the quantity of heat which the molecule has just acquired, we shall have its increase of temperature. Thus we obtain the general equation

which is the equation of the propagation of heat in the interior of all solid bodies.

143. Independently of this equation the system of temperatures is often subject to several definite conditions, of which no general expression can be given, since they depend on the nature of the problem.

If the dimensions of the mass in which heat is propagated are finite, and if the surface is maintained by some special cause in a given state; for example, if all its points retain, by virtue of that cause, the constant temperature 0, we shall have, denoting the unknown function v by p (x, y, z, t), the equation of condition. $ (x, y, z, t) = 0; which must be satisfied by all values of x, y, z which belong to points of the external surface, whatever be the value of t. Further, if we suppose the initial temperatures of the body to be expressed by the known function F(x, y, z), we have also the equation (x, y, z, 0) = F(x, y, z); the condition expressed by this equation must be fulfilled by all values of the co-ordinates x, y, z which belong to any point whatever of the solid.

144. Instead of submitting the surface of the body to a constant temperature, we may suppose the temperature not to be the same at different points of the surface, and that it varies with the time according to a given law; which is what takes place in the problem of terrestrial temperature. In this case the equation relative to the surface contains the variable t.

145. In order to examine by itself, and from a very general point of view, the problem of the propagation of heat, the solid whose initial state is given must be supposed to have all its dimensions infinite; no special condition disturbs then the dif

fusion of heat, and the law to which this principle is submitted becomes more manifest; it is expressed by the general equation

to which must be added that which relates to the initial arbitrary state of the solid.

Suppose the initial temperature of a molecule, whose coordinates are x, y, z, to be a known function F(x, y, z), and denote the unknown value v by 4 (x, y, z, t), we shall have the definite equation (x, y, z, 0) = F(x, y, z); thus the problem is reduced to the integration of the general equation (A) in such a manner that it may agree, when the time is zero, with the equation which contains the arbitrary function F.

SECTION VII.

General equation relative to the surface.

146. If the solid has a definite form, and if its original heat is dispersed gradually into atmospheric air maintained at a constant temperature, a third condition relative to the state of the surface must be added to the general equation (A) and to that which represents the initial state.

We proceed to examine, in the following articles, the nature of the equation which expresses this third condition.

Consider the variable state of a solid whose heat is dispersed into air, maintained at the fixed temperature 0. Let w be an infinitely small part of the external surface, and μ a point of w, through which a normal to the surface is drawn; different points of this line have at the same instant different temperatures.

Letv be the actual temperature of the point μ, taken at a definite instant, and w the corresponding temperature of a point v of the solid taken on the normal, and distant from μ by an infinitely small quantity a. Denote by x, y, z the co-ordinates of the point μ, and those of the point v by x+dx, y+dy, z+dz; let f(x, y, z) = 0 be the known equation to the surface of the solid, and v = (x, y, z, t) the general equation which ought to give the

value of v as a function of the four variables x, y, z, t. Differentiating the equation ƒ (x, y, z) = 0, we shall have

mdx+ndy+pdz = 0;

m, n, p being functions of x, y, z.

It follows from the corollary enunciated in Article 141, that the flow in direction of the normal, or the quantity of heat which during the instant dt would cross the surface w, if it were placed at any point whatever of this line, at right angles to its direction, is proportional to the quotient which is obtained by dividing the difference of temperature of two points infinitely near by their distance. Hence the expression for the flow at the end of the normal is

K denoting the specific conducibility of the mass. On the other hand, the surface o permits a quantity of heat to escape into the air, during the time dt, equal to hvwdt; h being the conducibility relative to atmospheric air. Thus the flow of heat at the end of the normal has two different expressions, that is to say:

hence these two quantities are equal; and it is by the expression of this equality that the condition relative to the surface is introduced into the analysis.

Now, it follows from the principles of geometry, that the coordinates dx, dy, Sz, which fix the position of the point v of the normal relative to the point μ, satisfy the following conditions: