4GO THEORY OF HEAT. [CHAP. IX. 



ture of a point is proportional to the excess of the quantity of heat 

 which that point receives over the quantity which it has lost, and 

 that a partial differential equation must express this result : but 

 the problem does not consist in enunciating this proposition which 

 is the mere fact; it consists in actually forming the differential 

 equation, which requires that we should consider the fact in its 

 elements. If instead of employing the exact expression of the 

 flow of heat, we omit the denominator of this expression, we 

 thereby introduce a difficulty which is nowise inherent in the 

 problem; there is no mathematical theory which would not offer 

 similar difficulties, if we began by altering the principle of the 

 proofs. Not only are we thus unable to form a differential equa 

 tion; but there is nothing more opposite to an equation than a 

 proposition of this kind, in Avhich we should be expressing the 

 equality of quantities which could not be compared. To avoid 

 this error, it is sufficient to give some attention to the demon 

 stration and the consequences of the foregoing lemma (Art. 65, 

 66, 67, and Art. 75). 



4th. With respect to the ideas from which we have deduced 

 for the first time the differential equations, they are those which 

 physicists have always admitted. We do not know that anyone 

 has been able to imagine the movement of heat as being produced 

 in the interior of bodies by the simple contact of the surfaces 

 which separate the different parts. For ourselves such a proposition 

 would appear to be void of all intelligible meaning. A surface of 

 contact cannot be the subject of any physical quality; it is neither 

 heated, nor coloured, nor heavy. It is evident that when one 

 part of a body gives its heat to another there are an infinity 

 of material points of the first which act on an infinity of points of 

 the second. It need only be added that in the interior of opaque 

 material, points whose distance is not very small cannot commu 

 nicate their heat directly; that which they send out is intercepted 

 by the intermediate molecules. The layers in contact are the only 

 ones which communicate their heat directly, when the thickness 

 of the layers equals or exceeds the distance which the heat sent 

 from a point passes over before being entirely absorbed. There is 

 no direct action except between material points extremely near, 

 and it is for this reason that the expression for the flow has the 

 form which we assign to it. The flow then results from an infinite 



