356 THEORY OF HEAT. [CHAP. IX. 



given by M. Laplace 1 , in a work which forms part of volume vui 

 of the Me moires de 1 Ecole Polytechnique ; we apply it simply to 

 the determination of the linear movement of heat. From it we 

 conclude 



,, f +0 , 2JL/ 



y g-hti dqe-V([&amp;gt;(x + 



J -00 



when t = the value of u is F(x) e 

 hence 



= r 



J _ 



and &amp;lt;&amp;gt; x = = 



Thus the arbitrary function which enters into the integral, is deter 

 mined by means of the given function /(a?), and we have the 

 following equation, which contains the solution of the problem, 



/WL e~ M f +0 , 



v = -^e * + -7=- dqe-^f (x + Sta/ftj) , . 



V 7T / _oo 



it is easy to represent this solution by a construction. 



365. Let us apply the previous solution to the case in which 

 all points of the line AB having the initial temperature 0, the end 

 A. is heated so as to be maintained continually at the tempera 

 ture 1. It follows from this that F (x) has a nul value when x 



-x !^~ L 

 differs from 0. Thus f(x} is equal to e KS whenever x differs 



from 0, and to when x is nothing. On the other hand it is 

 necessary that on making x negative, the value off(x) should change 

 sign, so that we have the condition /( x) f(x) We thus 

 know the nature of the discontinuous function f(x) t it becomes 



. - 



e when x exceeds 0, and + e KS when x is less than 0. 



We must now write instead of x the quantity x + 2q^kt. To find 



r +co vi 



u orl dqe-* -. f(x+ %VAtf), we must first take the integral 



from 



= to 



1 Journal de TEcole Polytechnique, Tome vm. pp. 235244, Paris, 1809. 

 Laplace shews also that the complete integral of the equation contains only one 

 arbitrary function, but in this respect he had been anticipated by Poisson. [A. F.J 



