M. POISSON ON THE MATHEMATICAL THEORY OF HEAT. 127 
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 through- 
out 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 con- 
sidered 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 ex- 
cluded 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. 
Cuapter 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 tempe- 
rature varies very rapidly. From this equality, and from the expression 
of the exterior radiation, given by observation, we determine the equa- 
tion of the motion of heat at the surface of a body of any form what- 
soever. The expression of the interior passage not being applicable to 
the surface itself, it follows that the demonstration of this general equa- 
tion, which consists in immediately equalizing that expression to the ex- 
pression 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 sur- 
face; they contain one quantity depending on the material of those two 
parts respectively, and which can only be determined by experiment. 
Cuaprer 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 
