362 THEORY OF HEAT. [CHAP. IX. 



=- or 



thus the heat acquired increases proportionally to the square root of 

 the time elapsed. 



367. By a similar analysis we may treat the problem of the 

 diffusion of heat, which also depends on the integration of the 



equation ~r: = k j-^ hv. Representing by f^x) the initial tem 

 perature of a point in the line situated at a distance x from the 

 origin, we proceed to determine what ought to be the temperature 

 of the same point after a time t. Making v = e~ ht z, we have 



-y- = k -Tg- , and consequently z I dq e~ qt ^&amp;gt; (x + 2q Jkt). When 



(it Ut J -oo 



t 0, we must have 



9 ( x ) or 



J GO 



hence 



e~ty 



To apply this general expression to the case in which a part of 

 the line from x ato# = ais uniformly heated, all the rest of 

 the solid being at the temperature 0, we must consider that the 

 factor f(x+ 2q Jfo) which multiplies e~ qZ has, according to hypo 

 thesis, a constant value 1, when the quantity which is under the 

 sign of the function is included between a and a, and that all 

 the other values of this factor are nothing. Hence the integral 



Idq e-v* ought to be taken from x+2q Jkt = a to x + 2q JTt = a, 



or from q= --j^.~ toq= . Denoting as above by -^ & (It) 



**jkt *&amp;gt;Jkt VTT 



the integral ldre~ rZ taken from r = R to r = oo , we have 



2jktn 



