SECT. II.] APPROXIMATE FORMULA. 383 



/g? J_ 7/ 2 _|_ 2 2 \ 



integration, reduces to f ^- J, when tlie value of the time 



increases without limit. In fact, the terms such as ^,- and ^r- 



receive in this case very small absolute values, since the numera 

 tors are included between fixed limits, and the denominators 

 increase to infinity. Thus the factors which we omit differ 

 extremely little from unity. Hence the variable state of the 

 solid, after a great value of the time, is expressed by 



The factor Idildft ldyf(z, /9, 7) represents the whole quantity 



of heat B which the solid contains. Thus the system of tempera 

 tures depends .not upon the initial distribution of heat, but only 

 on its quantity. We might suppose that all the initial heat was 

 contained in a single prismatic element situated at the origin, 

 whose extremely small orthogonal dimensions were a) lt &&amp;gt; 2 , o&amp;gt; 3 . The 

 initial temperature of this element would be denoted by an 

 exceedingly great number /, and all the other molecules of the 

 solid would have a nul initial temperature. The product 

 G) i ft) 2 Ct) 3/ i g equal in this case to the integral 



Whatever be the initial heating, the state of the solid which 

 corresponds to a very great value of the time, is the same as if all 

 the heat had been collected into a single element situated at the 



385. Suppose now that we consider only the points of the 

 solid whose distance from the origin is very great with respect 

 to the dimensions of the heated part ; we might first imagine 

 that this condition is sufficient to reduce the exponent of e in 

 the general integral. The exponent is in fact 



