and the lateral surface of which is supposed to be impermeable to heat and its two bases retained at constant temperatures, the passage of heat across every section perpendicular to its length is the same throughout its length. Its magnitude is proportional to the temperature of the two bases, and in the inverse ratio of the distance which separates them. This principle is easy to demonstrate, or rather it may be considered as evident. Thus expressed, it is independent of the mode of communication of heat, and it takes place whatever be the length of the prism: but it was erroneous to have attributed it without restriction to the infinitely thin slices of one body, the temperature of which varies, either with the time, or from one point to another; and to have excluded from it the circumstance, that the equation of the movement of heat, deduced from that of extension, is independent of any hypothesis and comparable in its generality to the theorems of statics. When we make no supposition respecting the mode of communication of heat, or the law of interior radiation, the passage of heat through each face of an infinitely thin slice is no longer simply proportional to the infinitely small difference of the temperatures of the two faces, or in the inverse ratio of the thickness of the slices; the exact expression of it will be found in the chapter in which we treat specially of the distribution of heat in a prismatic bar.
Chapter V. On the Movement of Heat at the Surface of a Body of any Form.—We demonstrate that the passages of heat are equal, or become so after a very short time, in the two extremities of a prism which has for its base an element of the surface of a body, and is in height a little greater than the thickness of the superficial layer, in which the temperature varies very rapidly. From this equality, and from the expression of the exterior radiation, given by observation, we determine the equation of the motion of heat at the surface of a body of any form whatsoever. The expression of the interior passage not being applicable to the surface itself, it follows that the demonstration of this general equation, which consists in immediately equalizing that expression to the expression of the exterior radiation, is altogether illusory.
When a body is composed of two parts of different materials, two equations of the motion of heat exist at their surface of separation, which are demonstrated in the same manner as the equation relative to the surface; they contain one quantity depending on the material of those two parts respectively, and which can only be determined by experiment.
Chapter VI. A Digression on the Integrals of Equations of partial Differences.—By the consideration of series, we demonstrate that the number of arbitrary constants contained in the complete integral of a differential equation ought always to be equal to that which indicates
