380 THEORY OF HEAT. [CHAP. IX. 



and that and g may be the limits of the portion originally 

 heated. 



The exact solution is given by the equation 



(a-*) 2 



r^a/(a)e- ** ,. 



1} = i , i A / , 



Jo Zirkt 



and the approximate solution is given by the equation 



(y), 



k denoting the value ^j^ of the conducibility. In order that the 

 equation (y) may be substituted for the preceding equation (i} ) it 



2ax-a? 



is in general requisite that the factor e *M , which is that which 

 we omit, should differ very little from unity ; for if it were 1 or \ 

 we might apprehend an error equal to the value calculated or to 



the half of that value. Let then e &* 1 + w, to being a small 



fraction, as ^^ or 77:7:7,; from this we derive the condition 

 LOO LOOO 



a 2 \ 

 I , 

 J 



= a&amp;gt;, or t 



co 



and if the greatest value g which the variable a can receive is 



1 O3C 



very small with respect to x, we have t = - ^y . 



co ^i/2 



We see by this result that the more distant from the origin 

 the points are whose temperatures we wish to determine by means 

 of the reduced equation, the more necessary it is for the value of 

 the time elapsed to be great. Thus the heat tends more and more 

 to be distributed according to a law independent of the primitive 

 heating. After a certain time, the diffusion is sensibly effected, 

 that is to say the state of the solid depends on nothing more than 

 the quantity of the initial heat, and not on the distribution which 

 was made of it. The temperatures of points sufficiently near to 

 the origin are soon represented without error by the reduced 

 equation (y}\ but it is not the same with points very distant from 



