SECT. II.] INITIAL HEAT COLLECTED AT THE ORIGIN. 377 



+ff 



dnf(d) denotes the whole quantity of heat B contained in the 



-h 



solid, and we see that the primitive distribution has no influence 

 on the temperatures after a very long time. They depend only 

 on the sum B, and not on the law according to which the heat has 

 been distributed. 



378. If we suppose a single element co situated at the origin 

 to have received the initial temperature/ and that all the others 

 had initially the temperature 0, the product cof will be equal to 



r+ff 

 the integral I &amp;lt;fa/(a) or B. The constant /is exceedingly great 



J h 



since we suppose the line co very small. 



X* 



The equation v = ._ .. cof represents the movement which 



2 J TT *Jt 



would take place, if a single element situated at the origin had 

 been heated. In fact, if we give to x any value a, not infinitely 



X 2 



small, the function - will be nothing when we suppose t = 0. 



The same would not be the case if the value of x were 



_- 

 nothing. In this case the function receives on the contrarv 



an infinite value when t = 0. We can ascertain distinctly the 

 nature of this function, if we apply the general principles of the 

 theory of curved surfaces to the surface whose equation is 



e~ty 



g 



The equation v = ._ ._ a)f expresses then the variable tem- 



perature at any point of the prism, when we suppose the whole 

 initial heat collected into a single element situated at the origin. 

 This hypothesis, although special, belongs to a general problem, 

 since after a sufficiently long time, the variable state of the solid is 

 always the same as if the initial heat had been collected at the 

 origin. The law according to which the heat was distributed, has 



