354 THEORY OF HEAT. [CHAP. IX. 



the final temperature, we must find for u a function which satisfies 



the equation -r^k-r^ hu, and whose initial value is f(x), and 

 ctt cl/x 



-x&amp;gt;J T ^ 



final value 0. At the point A, or x = 0, the quantity v-e 

 has, by hypothesis, a constant value equal to 0. We see by this 

 that u represents an excess of heat which is at first accumulated in 

 the prism, and which then escapes, either by being propagated to 

 infinity, or by being scattered into the medium. Thus to represent 

 the effect which results from the uniform heating of the end A of 

 a line infinitely prolonged, we must imagine, 1st, that the line is 

 also prolonged to the left of the point A, and that each point 

 situated to the right is now affected with the initial excess of 

 temperature ; 2nd, that the other half of the line to the left of 

 the point A is in a contrary state ; so that a point situated at the 

 distance - x from the point A has the initial temperature /(#) : 

 the heat then begins to move freely through the interior of the 

 bar, and to be scattered at the surface. 



The point A preserves the temperature 0, and all the other 

 points arrive insensibly at the same state. In this manner we are 

 able to refer the case in which the external source incessantly com 

 municates new heat, to that in which the primitive heat is propa 

 gated through the interior of the solid. We might therefore solve 

 the proposed problem in the same manner as that of the diffusion 

 of heat, Articles 347 and 353; but in order to multiply methods of 

 solution in a matter thus new, we shall employ the integral under 

 a different form from that which we have considered up to this 

 point. 



364. The equation -^ = k -7-3 is satisfied by supposing u equal 



to e~ x e kt . This function of x and t may also be put under the form 

 of a definite integral, which is very easily deduced from the known 



value of ldqe~ q \ We have in fact *j7r=]dqe~ q *, when the integral 

 is taken from = -coto = +oo. We have therefore also 



J JT \dqe~ 



