tained long since for heterogeneous bodies. The variations of temperature which take place at every instant, and arise from the mutual radiation of the neighbouring molecules, depend in fact only on their actual positions, and not at all on the positions in which they will be the instant after in consequence of the motions produced by their calorific action or by other causes: it is thus that in the problem of the flux and reflux of the tides, for example, we calculate the attraction of the sea upon each point of its mass, as if it were solid and at rest at the moment under consideration, and independently of the motions which this attraction may produce.
Notwithstanding that the interior radiation takes place only between molecules the temperatures of which are extremely different, the equation of motion of the heat contains terms derived from the squares of their differences, and of the same order of magnitude as those which result from their first power; so that the exact equation differs, in the case of a homogeneous body, from that which we had already given; and it is not, like that, independent of the conductibility when the body has arrived at an invariable state. This equation of partial differences changes its form, when we cannot consider the extent of the interior radiation as insensible; it is then of a higher order, which introduces, in its integral, new constants or arbitrary functions. From this a difficulty of analysis arises, of which we give the solution, and explain how in every case the redundant quantities will be made to disappear, as will be seen from a particular example in another chapter. We form in this the general expression of the passage of heat through every element of a surface traced in the interior of a body which is heated or cooled, or has arrived at an invariable state, and in which the extent of the interior radiation is considered as insensible. This passage proceeds from the exchange of heat between the molecules of the two parts of that body near their surface of separation, and the temperatures of which are very different; whilst the interior passage results from the exchanges between the molecules adjacent to the surface of the body and those of a surrounding medium, or of other bodies which may have much higher or much lower temperatures; and notwithstanding that the respective magnitudes of these two passages (ces deux flux), due to causes also unequal, must be of the same order and comparable with one another. We show how that condition is fulfilled, by means of a quantity resulting from the rapid decrease of temperature which takes place very near the surface of a body whilst heating or cooling. In this manner interior and exterior passages are found united with one another; and the law of interior conductibility expressed in functions of the temperature is deduced from that of exterior radiation which MM. Dulong and Petit have discovered.
In a homogeneous prism which has arrived at an invariable state,
