SECT. I.] VARYING TEMPERATURE IX IX FINITE BAR. 333 



the following article. We shall first examine the case in which 

 the source of heat is constant. 



Suppose that, the initial heat being distributed in any manner 

 throughout the infinite bar, we maintain the section A at a 

 constant temperature, whilst part of the heat communicated is dis 

 persed through the external surface. It is required to determine 

 the state of the prism after a given time, which is the object of the 

 second problem that we have proposed to ourselves. Denoting by 

 1 the constant temperature of the end A, by that of the medium, 



W^ 

 we have e S as the expression of the final temperature of a 



point situated at the distance x from this extremity, or simply 



TTJ- 



e~ x j assuming for simplicity the quantity - y to be equal to unity. 



Denoting by v the variable temperature of the same point after 

 the time t has elapsed, we have, to determine v, the equation 



dvct*v HL 



_ 



let now v = e~ Ks +u, 



du K d*a HL , 



vehftve 



dit , (TV 



- = k 



rr TT T 



replacing by k and by h. 



If we make u=e~ ht u we have -,- Jc j- a : the value of u or 



dt dx a 



W 



v e Ks is that of the difference between the actual and the 

 final temperatures ; this difference u, which tends more and more 

 to vanish, and whose final value is nothing, is equivalent at first to 



-W^ 



F(x)r-e *, 



denoting by F (x) the initial temperature of a point situated at the 



distance x. Let f(x) be the excess of the initial temperature over 



F. H. !:} 



