C2 THEORY OF HEAT. [CHAP. I. 



80. It is easy to ascertain how much heat flows during unit 

 of time through a section of the bar arrived at its fixed state : 



7 I2A 



this quantity is expressed by 4K 2 -y- , or kAjkhl*.e * K j and 



if we take its value at the origin, we shall have bAjZkh? as the 

 measure of the quantity of heat which passes from the source 

 into the solid during unit of time ; thus the expenditure of the 

 source of heat is, all other things being equal, proportional to the 

 square root of the cube of the thickness. 



We should obtain the same result on taking the integral 

 fShlv . dx from x nothing to x infinite. 



SECTION VI. 

 On the heating of closed spaces. 



81. We shall again make use of the theorems of Article 72 

 in the following problem, whose solution offers useful applications ; 

 it consists in determining the extent of the heating of closed 

 spaces. 



Imagine a closed space, of any form whatever, to be filled with 

 atmospheric air and closed on all sides, and that all parts of the 

 boundary are homogeneous and have a common thickness e, so 

 small that the ratio of the external surface to the internal surface 

 differs little from unity. The space which this boundary termi 

 nates is heated by a source whose action is constant ; for example, 

 by means of a surface whose area is cr maintained at a constant 

 temperature a. 



We consider here only the mean temperature of the air con 

 tained in the space, without regard to the unequal distribution of 

 heat in this mass of air ; thus we suppose that the existing causes 

 incessantly mingle all the portions of air, and make their tem 

 peratures uniform. 



We see first that the heat which continually leaves the source 

 spreads itself in the surrounding air and penetrates the mass of 

 which the boundary is formed, is partly dispersed at the surface, 



