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 perjiendicular to its length is the same tlirough- 

 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 m 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 

 Fcrrm. — We demonstrate that the passages of heat are equal, or become 

 so after a very short time, in the two extremities of a prism Avhich 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. 



Chapter VI. A Digression on the Integrals of Equations of partial 

 Differences. — By the consideration of series, we demonstrate that tiie 

 number of arbitrarj-^ constants contained in the complete integral of a 

 differential equation ought always to be equal to that which indicates 



