SECT. IV.] FORMATION OF EQUATIONS OF MOVEMENT. 457 



directly to the other during one instant a certain quantity of heat; 

 which quantity is proportional to the extremely small difference of 

 the temperatures: that is to say, if that difference became double, 

 triple, quadruple, and all other conditions remained the same, the 

 heat communicated would be double, triple, quadruple. 



This proposition expresses a general and constant fact, which 

 is sufficient to serve as the foundation of the mathematical theory. 

 The mode of transmission is then known with certainty, inde 

 pendently of every hypothesis on the nature of the cause, and 

 cannot be looked at from two different points of view. It is 

 evident that the direct transfer is effected in all directions, and 

 that it has no existence in fluids or liquids which are not diather- 

 manous, except between extremely near molecules. 



The general equations of the movement of heat, in the 

 interior of solids of any dimensions, and at the surface of these 

 bodies, are necessary consequences of the foregoing proposition. 

 They are rigorously derived from it, as we have proved in our 

 first Memoirs in 1807, and we easily obtain these equations by 

 means of lemmas, whose proof is not less exact than that of the 

 elementary propositions of mechanics. 



These equations are again derived from the same proposition, 

 by determining by means of integrations the whole quantity of 

 heat which one molecule receives from those which surround it. 

 This investigation is subject to no difficulty. The lemmas in 

 question take the place of the integrations, since they give directly 

 the expression of the flow, that is to say of the quantity of heat, 

 which crosses any section. Both calculations ought evidently to 

 lead to the same result; and since there is no difference in the 

 principle, there cannot be any difference in the consequences. 



2nd. We gave in 1811 the general equation relative to the 

 surface. It has not been deduced from particular cases, as has 

 been supposed without any foundation, and it could not be; the 

 proposition which it expresses is not of a nature to be discovered 

 by way of induction; we cannot ascertain it for certain bodies and 

 ignore it for others; it is necessary for all, in order that the state 

 of the surface may not suffer in a definite time an infinite change. 

 In our Memoir we have omitted the details of the proof, since 



