388 THEORY OF HEAT. [CHAP. IX. 



ture of the heated layer will therefore be , and from this follows 



CO 



what was said before, that the variable state of the solid is 

 expressed by the equation 



fb e^t 

 - J ~~ ht (a) ; 



this result holds when the coefficient -^ which enters into the 



L/JJ 



differential equation -=- = -^= -^ z hv, is denoted by k. As to the 



777 



coefficient h, it is equal to / ^ rtc/ ; S denoting the area of the 



section of the prism, I the contour of that section, and H the 

 conducibility of the external surface. 



Substituting these values in the equation (a) we have 



f represents the mean initial temperature, that is to say, that 

 which a single point would have if the initial heat were distributed 

 equally between the points of a portion of the bar whose length 

 is /, or more simply, unit of measure. It is required to determine 

 the value t of the time elapsed, which corresponds to a maximum 

 of temperature at a given point. 



To solve this problem, it is sufficient to derive from equation 

 (a) the value of -7- , and equate it to zero ; we have 

 dv , x* lv 



hence the value 0, of the time which must elapse in order that the 

 point situated at the distance x may attain its highest temperature, 

 is expressed by the equation 



