3~82 THEORY OF HEAT. [CHAP. IX. 



383. It follows from this examination that we ought not to 



1 W _(a-ff) 



conclude from the integral v = 7= &amp;lt;fe/(a) e ~4* &quot; that the 



law of the primitive distribution has no influence on the tempera 

 ture of points very distant from the origin. The resultant effect 

 of this distribution soon ceases to have influence on the points 

 near to the heated portion; that is to say their temperature 

 depends on nothing more than the quantity of the initial heat, 

 and not on the distribution which was made of it : but greatness 

 of distance does not concur to efface the impress of the distribu 

 tion, it preserves it on the contrary during a very long time 

 and retards the diffusion of heat. Thus the equation 



only after an immense time represents the temperatures of points 

 extremely remote from the heated part. If we applied it without 

 this condition, we should find results double or triple of the true 

 results, or even incomparably greater or smaller; and this would 

 not only occur for very small values of the time, but for great 

 values, such as an hour, a day, a year. Lastly this expression 

 would be so much the less exact, all other things being equal, as 

 the points were more distant from the part originally heated. 



384. When the diffusion of heat is effected in all directions, 

 the state of the solid is represented as we have seen by the 

 integral 



If the initial heat is contained in a definite portion of the solid 

 mass, we know the limits which comprise this heated part, and 

 the quantities a, /3, 7, which vary under the integral sign, cannot 

 receive values which exceed those limits. Suppose then that we 

 mark on the three axes six points whose distances are + X, + Y f +Z, 

 and X, Y, Z, and that we consider the successive states of 

 the solid included within the six planes which cross the axes at 

 these distances; we see that the exponent of e under the sign of 



