SECT. I.] COOLING OF AN INFINITE BAR. 363 



368. We shall next apply the general equation 



7T J 



to the case in which the infinite bar, heated by a source of 

 constant intensity 1, has arrived at fixed temperatures and is 

 then cooling freely in a medium maintained at the temperature 

 0. For this purpose it is sufficient to remark that the initial 



_ X J* 

 function denoted by f(x) is equivalent to e v * so long as the 



variable x which is under the sign of the function is positive, 



and that the same function is equivalent to e^* when the 

 variable which is affected by the symbol /is less than 0. Hence 



the first integral must be taken from 



x + 2q-Jkt = to x + fy-Jkt = oo , 

 and the second from 



x + ZqjTtt - - oo to x + 207^ = 0. 

 The first part of the value of v is 



e~ht fie r _ 



. Q-X\ jfc&quot; {(JqQ ^Q ^fl^Jht 



Jv J 



or 



or ^ .&quot;[dre-** , 

 making r = g 4- ^/Ai. The integral should be taken from 

 2 = ^r to 2 = &amp;gt; 



or from r = = to r 



