SECT. III.] GENERAL INVESTIGATION. 393 



which must elapse before this last effect is set up is exceedingly 

 great when the points of the mass are very distant from the origin. 

 Each of these points which had at first the temperature is 

 imperceptibly heated; its temperature then acquires the greatest 

 value which it can receive; and it ends by diminishing more and 

 more, until there remains no sensible heat in the mass. The 

 variable state is in general represented by the equation 



V =fdajdbfdo e - ^ -/(o,M ......... (E). 



The integrals must be taken between the limits 



The limits a lt + a 2 , b lt + b 2 , c 1 , + c 2 are given; they 

 include the whole portion of the solid which was originally heated. 

 The function f(a, b, c) is also given. It expresses the initial 

 temperature of a point whose co-ordinates are a, b, c. The defi 

 nite integrations make the variables a, b, c disappear, and there 

 remains for v a function of x, y, z, t and constants. To determine 

 the time which corresponds to a maximum of v, at a given point 



ra, we must derive from the preceding equation the value of -57: 



at 



we thus form an equation which contains 6 and the co-ordinates of 

 the point ra. From this we can then deduce the value of 6. If 

 then we substitute this value of 6 instead of t in equation (E), we 

 find the value of the highest temperature V expressed in x } y } z 

 and constants. 



Instead of equation (E) let us write 



v = (da fdb jdc Pf(a, b, c), 

 denoting by P the multiplier of f (a, b, c), we have 



dt = ~2 t+j da db ) dc gs 



393. We must now apply the last expression to points of the 

 solid which are very distant from the origin. Any point what 

 ever of the portion which contains the initial heat, having for co 

 ordinates the variables a, b, c, and the co-ordinates of the point m 



