SECT. V.] STEADY TEMPERATURE IX A BAR. 59 



and the second a lower temperature v , we see that the quantity 

 of heat which it receives through the first surface depends on 

 the difference v v , and is proportional to it : but this remark 

 is not sufficient to complete the calculation. The quantity in 

 question is not a differential : it has a finite value, since it is 

 equivalent to all the heat which escapes through that part of 

 the external surface of the prism which is situate to the right 

 of the section. To form an exact idea of it, we must compare 

 the lamina whose thickness is dx, with a solid terminated by 

 two parallel planes whose distance is e, and which are maintained 

 at unequal temperatures a and b. The quantity of heat which 

 passes into such a prism across the hottest surface, is in fact 

 proportional to the difference a b of the extreme temperatures, 

 but it does not depend only on this difference : all other things 

 being equal, it is less when the prism is thicker, and in general 



it is proportional to . This is why the quantity of heat 



^ 



which passes through the first surface into the lamina, whose 



thickness is dx } is proportional to -= . 



dx 



We lay stress on this remark because the neglect of it has 

 been the first obstacle to the establishment of the theory. If 

 we did not make a complete analysis of the elements of the 

 problem, we should obtain an equation not homogeneous, and, 

 a fortiori, we should not be able to form the equations which 

 express the movement of heat in more complex cases. 



It was necessary also to introduce into the calculation the 

 dimensions of the prism, in order that we might not regard, as 

 general, consequences which observation had furnished in a par 

 ticular case. Thus, it was discovered by experiment that a bar 

 of iron, heated at one extremity, could not acquire, at a distance 

 of six feet from the source, a temperature of one degree (octo- 

 gesimal 1 ) ; for to produce this effect, it would be necessary for 

 the heat of the source to surpass considerably the point of fusion 

 of iron; but this result depends on the thickness of the prism* 

 employed. If it had been greater, the heat would have been, 

 propagated to a greater distance, that is to say, the point of 

 the bar which acquires a fixed temperature of one degree is 

 1 Reaumur s Scale of Temperature. [A. F.J 



