464 THEORY OF HEAT. [CHAP. IX. 



the usefulness of it, especially when the analysis consists of de 

 finite integrals, and the limits of the integrals are themselves the 

 object of investigation. 



432. The chief results of our theory are the differential equa 

 tions of the movement of heat in solid or liquid bodies, and the 

 general equation which relates to the surface. The truth of these 

 equations is not founded on any physical explanation of the effects 

 of heat. In whatever manner we please to imagine the nature of 

 this element, whether we regard it as a distinct material thing 

 which passes from one part of space to another, or whether we 

 make heat consist simply in the transfer of motion, we shall always 

 arrive at the same equations, since the hypothesis which we form 

 must represent the general and simple facts from which the 

 mathematical laws are derived. 



The quantity of heat transmitted by two molecules whose 

 temperatures are unequal, depends on the difference of these 

 temperatures. If the difference is infinitely small it is certain 

 that the heat communicated is proportional to that difference ; all 

 experiment concurs in rigorously proving this proposition. Now 

 in order to establish the differential equations in question, we 

 consider only the reciprocal action of molecules infinitely near. 

 There is therefore no uncertainty about the form of the equations 

 which relate to the interior of the mass. 



The equation relative to the surface expresses, as we have said, 

 that the flow of the heat, in the direction of the normal at the 

 boundary of the solid, must have the same value, whether we cal 

 culate the mutual action of the molecules of the solid, or whether 

 we consider the action which the medium exerts upon the envelope. 

 The analytical expression of the former value is very simple and 

 is exactly known ; as to the latter value, it is sensibly proportional 

 to the temperature of the surface, when the excess of this tempera 

 ture over that of the medium is a sufficiently small quantity. In 

 other cases the second value must be regarded as given by a series 

 of observations; it depends on the surface, on the pressure and 

 on the nature of the medium ; this observed value ought to form 

 the second member of the equation relative to the surface. 



In several important problems, the equation last named is re- 



