SECT. III.] TIMES OF HIGHEST TEMPERATURES. 387 



which gives 



(x+V)* C*j^) 2 



(%+g}6~ ~v = (x-g}e~ ; 

 whence we conclude 



J 



We have supposed - rrf ^ = \. To restore the coefficient we 



Kt 



must write -^ instead of t, and we have 



__ ff CD x 



~K~r 



The highest temperatures follow each other according to the 

 law expressed by this equation. If we suppose it to represent the 

 varying motion of a body which describes a straight line, x being 

 the space passed over, and t the time elapsed, the velocity of 

 the moving body will be that of the maximum of temperature. 



When the quantity g is infinitely small, that is to say when the 

 initial heat is collected into a single element situated at the 



origin, the value of t is reduced to - , and by differentiation or 



, Kt x* 



development in series we find -^ = . 



(jD 2&amp;gt; 



We have left out of consideration the quantity of heat which 

 escapes at the surface of the prism ; w T e now proceed to take account 

 of that loss, and we shall suppose the initial heat to be contained 

 in a single element of the infinite prismatic bar. 



388. In the preceding problem we have determined the 

 variable state of an infinite prism a definite portion of which was 

 affected throughout with an initial temperature f. We suppose 

 that the initial heat was distributed through a finite space from 

 x = to x = b. 



We now suppose that the same quantity of heat If is contained 

 in an infinitely small element, from x = to x = a). The tempera- 



252 



