SECT. IV.] FLOW OUTWARD AND INTERNAL. 461 



multitude of actions whose effects are added ; but it is not from 

 this cause that its value during unit of time is a finite and 

 measurable magnitude, even although it be determined only by 

 an extremely small difference between the temperatures. 



When a heated body loses its heat in an elastic medium, or in 

 a space free from air bounded by a solid envelope, the value of the 

 outward flow is assuredly an integral; it again is due to the action 

 of an infinity of material points, very near to the surface, and we 

 have proved formerly that this concourse determines the law of 

 the external radiation 1 . But the quantity of heat emitted during 

 the unit of time would be infinitely small, if the difference of the 

 temperatures had not a finite value. 



In the interior of masses the conductive power is incomparably 

 greater than that which is exerted at the surface. This property, 

 whatever be the cause of it, is most distinctly perceived by us, 

 since, when the prism has arrived at its constant state, the 

 quantity of heat which crosses a section during the unit of time 

 exactly balances that which is lost through the whole part of the 

 heated surface, situated beyond that section, whose temperatures 

 exceed that of the medium by a finite magnitude. When we take 

 no account of this primary fact, and omit the divisor in the 

 expression for the flow, it is quite impossible to form the differen 

 tial equation, even for the simplest case; a fortiori, we should be 

 stopped in the investigation of the general equations. 



5th. Farther, it is necessary to know what is the influence of 

 the dimensions of the section of the prism on the values of the 

 acquired temperatures. Even although the problem is only that 

 of the linear movement, and all points of a section are regarded 

 as having the same temperature, it does not follow that we can 

 disregard the dimensions of the section, and extend to other prisms 

 the consequences which belong to one prism only. The exact 

 equation cannot be formed without expressing the relation 

 between the extent of the section and the effect produced at the 

 extremity of the prism. 



We shall not develope further the examination of the principles 

 which have led us to the knowledge of the differential equations ; 



1 Memoires de VAcadcmie des Sciences, Tome v. pp. 2048. Communicated 

 in 1811. [A. F.] 



