386 THEORY OF HEAT. [CHAP. IX. 



be the time t which corresponds to the maximum of temperature 

 at a given point whose distance is x. 



We have seen, in the preceding Articles, that the variable 

 temperature at any point is expressed by the equation 



f-*p 



-FT- 



The coefficient k represents -^n ^ being the specific con- 



Ox/ 



ducibility, C the capacity for heat, and D the density. 



To simplify the investigation, make Jc = 1, and in the result 



Tpi 



write kt or - instead of t. The expression for v becomes 



7 72 J7 



This is the integral of the equation -=- = -y- . The function -y- 



cfa oar cfo; 



measures the velocity with which the heat flows along the axis of 

 the prism. Now this value of -y-- is given in the actual problem 

 without any integral sign. We have in fact 



a x _! 



p 



or, effecting the integration, 



^=_/_ 



dx 2 



387. The function ~ z may also be expressed without the 



(ll) 



sign of integration: now it is equal to a fluxion of the first order^-; 

 hence on equating to zero this value of -=- , which measures the 



Uit 



instantaneous increase of the temperature at any point, we have 

 the relation sought between x and t. We thus find 



- 2 (* + ff~) f -^ , 2 ( - 



~ 



