SECT. III.] HIGHEST TEMPERATURES IN A SOLID. 385 



When we express analytically the temperature of these point?, 

 the object of the investigation is not to determine numerically 

 these temperatures, which are not measurable, but to ascertain 

 their ratios. Now these quantities depend certainly on the law 

 according to which the initial heat has been distributed, and the 

 effect of this initial distribution lasts so much the longer as the 

 parts of the prism are more distant from the source. But if the 



terms which form part of the exponent, such as -rj- and -7-7-, have 



4kt 4*kt 



absolute values decreasing without limit, we may employ the 

 approximate integrals. 



This condition occurs in problems where it is proposed to 

 determine the highest temperatures of points very distant from 

 the origin. We can demonstrate in fact that in this case the 

 values of the times increase in a greater ratio than the distances, 

 and are proportional to the squares of these distances, when the 

 points we are considering are very remote from the origin. It is 

 only after having established this proposition that we can effect 

 the reduction under the exponent. Problems of this kind are the 

 object of the following section. 



SECTION III. 



Of the highest temperatures in an infinite solid. 



386. We shall consider in the first place the linear move 

 ment in an infinite bar, a portion of which has been uniformly 

 heated, and we shall investigate the value of the time which must 

 elapse in order that a given point of the line may attain its 

 highest temperature. 



Let us denote by 2g the extent of the part heated, the middle 

 of which corresponds with the origin of the distances x. All the 

 points whose distance from the axis of y is less than g and greater 

 than g, have by hypothesis a common initial temperature f, and 

 all other sections have the initial temperature 0. We suppose 

 that no loss of heat occurs at the external surface of the prism, or, 

 which is the same thing, we assign to the section perpendicular to 

 the axis infinite dimensions. It is required to ascertain what will 

 F. H. 25 



