126 M. POISSON ON THE MATHEMATICAL THEORY OF HEAT- 



tained long since for hetei'ogeneous bodies. The variations of tem- 

 perature 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 ]3roblem 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 diff'erent, 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 par- 

 tial 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 

 everj^ 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 pas- 

 sage proceeds from the exchange of heat between the molecules of the 

 two parts of that body near their surface of separation, and the tempe- 

 ratures 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 nuich higher or much lower temperatures ; and notwithstand- 

 ing that the respective magnitudes of these two passages {ces deux flux), 

 due to causes also unequal, must be of the same order and com- 

 parable 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 

 wliich MM. Dulong and Petit have discovered. 



In a homogeneous prism which has arrived at an invariable state. 



