48 THEORY OF HEAT. [CHAP. I. 



result v v = &quot;&quot; a (z z). We shall then find, by the sub- 



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stitution of z + % and z + f, that the excess of temperature of 

 the point n over that of the point ri is also expressed by 



Z&amp;gt; a , 



It follows from this that the quantity of heat sent by the 

 point m to the point m will be the same as the quantity of heat 

 sent by the point n to the point ri, for all the elements which 

 concur in determining this quantity of transmitted heat are the 

 same. 



It is manifest that we can apply the same reasoning to every 

 system of two molecules which communicate heat to each other 

 across the section A or the section B f ; whence, if we could 

 sum up the whole quantity of heat which flows, during the same 

 instant, across the section A or the section J9 , we should find 

 this quantity to be the same for both sections. 



From this it follows that the part of the solid included be 

 tween A f and B receives always as much heat as it loses, and 

 since this result is applicable to any portion whatever of the 

 mass included between two parallel sections, it is evident that 

 no part of the solid can acquire a temperature higher than that 

 which it has at present. Thus, it has been rigorously demon 

 strated that the state of the prism will continue to exist just as it 

 was at first. 



Hence, the permanent temperatures of different sections of a 

 solid enclosed between two parallel infinite planes, are represented 

 by the ordinates of a straight line a/3, and satisfy the linear 



b a 

 equation v = a -\ --- z. 



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66. By what precedes we see distinctly what constitutes 

 the propagation of heat in a solid enclosed between two parallel 

 and infinite planes, each of which is maintained at a constant 

 temperature. Heat penetrates the mass gradually across the 

 lower plane : the temperatures of the intermediate sections are 

 raised, but can never exceed nor even quite attain a certain 

 limit which they approach nearer and nearer : this limit or final 

 temperature is different for different intermediate layers, and 



