302 THEORY OF HEAT. [CHAP. IX. 



We could then omit the term -=?- x~ } and we should have 1 



K 9 



In this case the value of the maximum is inversely propor 

 tional to the distance. Thus the movement of the wave would 

 not be uniform. It must be remarked that this hypothesis is 

 purely theoretical, and if the conducibility H is not nothing, but 

 only an extremely small quantity, the velocity of the wave is not 

 variable in the parts of the prism which are very distant from the 

 origin. In fact, whatever be the value of H t if this value is given, 

 as also those of K and g, and if we suppose that the distance x 



211 



increases without limit, the term -~r x z will always become much 



&9 

 greater than J. The distances may at first be small enough for 



2H 



the term -=- # 2 to be omitted under the radical. The times are 

 A# 



then proportional to the squares of the distances ; but as the heat 

 flows in direction of the infinite length, the law of propagation 

 alters, and the times become proportional to the distances. The 

 initial law, that is to say, that which relates to points extremely 

 near. to the source, differs very much from the final law which is 

 established in the very distant parts, and up to infinity : but, in 

 the intermediate portions, the highest temperatures follow each 

 other according to a mixed law expressed by the two preceding 

 equations (D) and ((7), 



392. It remains for us to determine the highest temperatures 

 for the case in which heat is propagated to infinity in every direc 

 tion within the material solid. This investigation, in accordance 

 with the principles which we have established, presents no 

 difficulty. 



When a definite portion of an infinite solid has been heated, 

 and all other parts of the mass have the same initial temperature 0, 

 heat is propagated in all directions, and after a certain time the 

 state of the solid is the same as if the heat had been originally 

 collected in a single point at the origin of co-ordinates. The time 



1 See equations (D) and (C), article 388, making 6 = 1. [A. F.] 



