126 M. POISSON ON THE MATHEMATICAL THEORY OF HEAT: 
tained long since for heterogeneous 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 eonsequence 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 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 
every element of a surface traced in the interior of a body which is 
heated or cooled, orhas 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 much 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 
which MM. Dulong and Petit have discovered. : 
In a homogeneous prism which has arrived at an invariable state, 
