SECT. IV.] UNIFORM LINEAR MOVEMENT. 53 



70. In order to establish the equations which we have 

 cited in Article 68, it would not -be necessary to suppose the 

 points which exert their action across the planes to be at ex 

 tremely small distances. 



^ The results would still be the same if the distances of these 

 points had any magnitude whatever ; they would therefore apply 

 also to the case where the direct action of heat extended within 

 the interior of the mass to very considerable distances, all the 

 circumstances which constitute the hypothesis remaining in other 

 respects the same. 



We need only suppose that the cause which maintains the 

 temperatures at the surface of the solid, affects not only that 

 part of the mass which is extremely near to the surface, but that 



its action extends to a finite depth. The equation V = a - a ~ b 2 



e 



will still represent in this case the permanent temperatures of 

 the solid. The true sense of this proposition is that, if we give 

 to all points of the mass the temperatures expressed by the 

 equation, and if besides any cause whatever, acting on the two 

 extreme laminae, retained always every one of their molecules 

 at the temperature which the same equation assigns to them, 

 the interior points of the solid would preserve without any change 

 their initial state. 



If we supposed that the action of a point of the mass could 

 extend to a finite distance e, it would be necessary that the 

 thickness of the extreme laminae, whose state is maintained by 

 the external cause, should be at least equal to e. But the 

 quantity e having in fact, in the natural state of solids, only 

 an inappreciable value, we may make abstraction of this thick 

 ness; and it is sufficient for the external cause to act on each 

 of the two layers, extremely thin, which bound the solid. This 

 is always what must be understood by the expression, to maintain 

 the temperature of the surface constant. 



71. We proceed further to examine the case in which the 

 same solid would be exposed, at one of its faces, to atmospheric 

 air maintained at a constant temperature. 



Suppose then that the lower plane preserves the fixed tem 

 perature a, by virtue of any external cause whatever, and that 



