58 THEORY OF HEAT. [CHAP. I. 



does not vary throughout the whole extent of the same section. 

 The thickness of the solid is dx, and the area of the section is 

 4/ 2 : hence the quantity of heat which flows uniformly, during 

 unit of time, across a section of this solid, is, according to the 



preceding principles, 4Z 2 A -=- , k being the specific internal con- 

 ducibility : we must therefore have the equation 



V&quot; 



whence 



^ \\\ i 



74. We should obtain the same result by considering the 

 equilibrium of heat in a single lamina infinitely thin, enclosed 

 between two sections at distances x arid x + dx. In fact, the 

 quantity of heat which, during unit of time, crosses the first 



section situate at distance x, is 4/ 2 X -r- . To find that which 



flows during the same time across the successive section situate 

 at distance x + dx, we must in the preceding expression change x 



into x + dx, which gives 4Z 2 &. ^~ + d ~ . If we subtract 



[dx \dxjj 



the second expression from the first we shall find how much 

 heat is acquired by the lamina bounded by these two sections 

 during unit of time ; and since the state of the lamina is per 

 manent, it follows that all the heat acquired is dispersed into 

 the air across the external surface Sldx of the same lamina : now 

 the last quantity of heat is Shlvdx : we shall obtain therefore the 

 same equation 



07 7 7 ^727 7 A&A 1 ^V 27?, 



8/uvdx klkd -y- , whence -^5 = -=-= v. 

 \dxj dx 2 kl 



75. In whatever manner this equation is formed, it is 

 necessary to remark that the quantity of heat which passes into 

 the lamina whose thickness is dx, has a finite value, and that 



its exact expression is 4&amp;lt;l 2 k ^- . The lamina being enclosed 

 between two surfaces the first of which has a temperature v, 



