intermediate problems, such as the problem of the linear movement of heat in a prism whose ends are maintained at fixed temperatures, or exposed to the atmospheric air. The expression for the varied movement of heat in a cube or rectangular prism which is cooling in an aëriform medium might be generalised, and any initial state whatever supposed. These investigations require no other principles than those which have been explained in this work. A memoir was published by M. Fourier in the Mémoires de l'Académie des Sciences, Tome VII. Paris, 1827, pp. 605-624, entitled, Mémoire sur la distinction des racines imaginaires, et sur l'application des théorèmes d'analyse algébrique aux équations transcendantes qui dependent de la théorie de la chaleur. It contains a proof of two propositions in the theory of heat. If there be two solid bodies of similar convex forms, such that corresponding elements have the same density, specific capacity for heat, and conductivity, and the same initial distribution of temperature, the condition of the two bodies will always be the same after times which are as the squares of the dimensions, when, 1st, corresponding elements of the surfaces are maintained at constant temperatures, or 2nd, when the temperatures of the exterior medium at corresponding points of the surface remain constant. : 8 For the velocities of flow along lines of flow across the terminal areas s, s' of corresponding prismatic elements are as u-v: u'-v′, where (u, v), (u, v) are temperatures at pairs of points at the same distance▲ on opposite sides of s and s'; and if n n' is the ratio of the dimensions, u - v : u' - v'n':n. If then, dt, dt' be corresponding times, 'the quantities of heat received by the prismatic elements are as sk (u – v) dt: s'k (u' – v') dt', or as n2n'dt: n2ndt'. But the volumes being as n3: n', if the corresponding changes of temperature are always equal we must have n2n'dt n'2ndt' dt n2 In the second case we must suppose H: H'=n': n. [A. F.] CHAPTER IX. OF THE DIFFUSION OF HEAT. FIRST SECTION. Of the free movement of heat in an infinite line. 342. HERE we consider the movement of heat in a solid homogeneous mass, all of whose dimensions are infinite. The solid is divided by planes infinitely near and perpendicular to a common axis; and it is first supposed that one part only of the solid has been heated, that, namely, which is enclosed between two parallel planes A and B, whose distance is g; all other parts have the initial temperature 0; but any plane included between A and B has a given initial temperature, regarded as arbitrary, and common to every point of the plane; the temperature is dif ferent for different planes. The initial state of the mass being thus defined, it is required to determine by analysis all the succeeding states. The movement in question is simply linear, and in direction of the axis of the plane; for it is evident that there can be no transfer of heat in any plane perpendicular to the axis, since the initial temperature at every point in the plane is the same. Instead of the infinite solid we may suppose a prism of very small thickness, whose lateral surface is wholly impenetrable to heat. The movement is then considered only in the infinite line which is the common axis of all the sectional planes of the prism. The problem is more general, when we attribute temperatures entirely arbitrary to all points of the part of the solid which has been heated, all other points of the solid having the initial temperature 0. The laws of the distribution of heat in an infinite solid mass ought to have a simple and remarkable character; since the movement is not disturbed by the obstacle of surfaces, or by the action of a medium. 343. The position of each point being referred to three rectangular axes, on which we measure the co-ordinates x, y, z, the temperature sought is a function of the variables x, y, z, and of the time t. This function v or p(x, y, z, t) satisfies the general equation dv K d2v d'v d'v = + + .(a). Further, it must necessarily represent the initial state which is arbitrary; thus, denoting by F(x, y, z) the given value of the temperature at any point, taken when the time is nothing, that is to say, at the moment when the diffusion begins, we must have Hence we must find a function v of the four variables x, y, z, t, which satisfies the differential equation (a) and the definite equation (b). In the problems which we previously discussed, the integral is subject to a third condition which depends on the state of the surface for which reason the analysis is more complex, and the solution requires the employment of exponential terms. The form of the integral is very much more simple, when it need only satisfy the initial state; and it would be easy to determine at once the movement of heat in three dimensions. But in order to explain this part of the theory, and to ascertain according to what law the diffusion is effected, it is preferable to consider first the linear movement, resolving it into the two following problems: we shall see in the sequel how they are applied to the case of three dimensions. 344. First problem: a part ab of an infinite line is raised at all points to the temperature 1; the other points of the line are at the actual temperature 0; it is assumed that the heat cannot be dispersed into the surrounding medium; we have to determine what is the state of the line after a given time. This problem may be made more general, by supposing, 1st, that the initial temperatures of the points included between a and b are unequal and represented by the ordinates of any line whatever, which we shall regard first as composed of two symmetrical parts (see fig. 16); 2nd, that part of the heat is dispersed through the surface of the solid, which is a prism of very small thickness, and of infinite length. The second problem consists in determining the successive states of a prismatic bar, infinite in length, one extremity of which is submitted to a constant temperature. The solution of these two problems depends on the integration of the equation (Article 105), which expresses the linear movement of heat. v is the temperature which the point at distance from the origin must have after the lapse of the time t; K, H, C, D, L, S, denote the internal and surface conducibilities, the specific capacity for heat, the density, the contour of the perpendicular section, and the area of this section. 345. Consider in the first instance the case in which heat is propagated freely in an infinite line, one part of which ab has received any initial temperatures; all other points having the initial temperature 0. If at each point of the bar we raise the ordinate of a plane curve so as to represent the actual temperature at that point, we see that after a certain value of the time t, the state of the solid is expressed by the form of the curve. Denote by v = F(x) the equation which corresponds to the given initial state, and first, for the sake of making the investigation been heated, all other points of the solid having the initial temperature 0. The laws of the distribution of heat in an infinite solid mass ought to have a simple and remarkable character; since the movement is not disturbed by the obstacle of surfaces, or by the action of a medium. 343. The position of each point being referred to three rectangular axes, on which we measure the co-ordinates x, y, z, the temperature sought is a function of the variables x, y, z, and of the time t. This function v or p(x, y, z, t) satisfies the general equation Further, it must necessarily represent the initial state which is arbitrary; thus, denoting by F(x, y, z) the given value of the temperature at any point, taken when the time is nothing, that is to say, at the moment when the diffusion begins, we must have $(x, y, z, 0) = F(x, y, z) ........................................ ..(b). Hence we must find a function v of the four variables x, y, z, t, which satisfies the differential equation (a) and the definite equation (b). In the problems which we previously discussed, the integral is subject to a third condition which depends on the state of the surface for which reason the analysis is more complex, and the solution requires the employment of exponential terms. The form of the integral is very much more simple, when it need only satisfy the initial state; and it would be easy to determine at once the movement of heat in three dimensions. But in order to explain this part of the theory, and to ascertain according to what law the diffusion is effected, it is preferable to consider first the linear movement, resolving it into the two following problems: we shall see in the sequel how they are applied to the case of three dimensions. 344. First problem: a part ab of an infinite line is raised at all points to the temperature 1; the other points of the line are at the actual temperature 0; it is assumed that the heat cannot be ispersed into the surrounding medium; we have to determine |