SECT. III.] CONDITIONS FOR DISTANT POINTS. 395 



product of several factors, such as 



-a 2 2 ax -x* 



Now it is not sufficient for the ratio - to be always a very 



great number in order that we may suppress the two first factors. 

 If, for example, we suppose a equal to a decimetre, and x equal to 

 ten metres, and if the substance in which the heat is propagated is 

 iron, we see that after nine or ten hours have elapsed, the factor 



2 ax 

 7 . 



e CD is still greater than 2 ; hence by suppressing it we should 



reduce the result sought to half its value. Thus the value of -r- , 



dt 



as it belongs to points very distant from the origin, and for any 

 time whatever, ought to be expressed by equation (a). But it is 

 not the same if we consider only extremely large values of the 

 time, which increase in proportion to the squares of the distances : 

 in accordance with this condition we must omit, even under the 

 exponential symbol, the terms which contain a, b, or c. Now this 

 condition holds when we wish to determine the highest tempera 

 ture which a distant point can acquire, as we proceed to prove. 



395. The value of ^- must in fact be nothing in the case in 

 question ; we have therefore 



Thus the time which must elapse in order that a very distant 

 point may acquire its highest temperature is proportional to the 

 square of the distance of this point from the origin. 



If in the expression for v we replace the denominator -^=- 



VjU 



2 

 by its value r 2 , the exponent of e~ l which is 



