CONSTITUTION OF MATTEE AND ANALTTICAL THEORIES DP HEAT. 



63 



_Hm '^(^v + Q-^(ü _ -2f\<o'{x',Q\^dx'. 

 Therefore J{t^) never increases as increases. 



Now consider lim J (Q. IJsing the notation of discontinuous analysis, 

 J {Q = (1, Q dj = [a, (1, Q m t,)] - / I cd' {X, Q I Jil Q dj, (7) 



Ä ^ _ 



where I t,) stands for / a (^', g c?!'. But , smce the initial temperature is 



— Tl 



zero , it follows from (A) that , for any arbitrarüy small but fixed quantity ^, 

 it is always possible to find a positive quantity i^^ such that 



whatever % may be. Tberefore, according to iii. of Art. 57, it follows from 

 (7) tbat 



p Standing for the greater of the two quantities which are the values 



of p for Vj, v^. Therefore lim J ify) = 0- Therefore J {t^ can never de- 



crease vpith the increase of i^,,, for it can never be negative. Therefore J {t^) 

 remains equal to zero vphatever t^. may be. But, according to i. of Art. 57, J (t^) 

 is a continuous function of and J[t) is the limital function corresponding to 

 it. Therefore J {t) is zero vv^hatever t may be. And, since according to i. of 

 Art. 57, CO {x, t) is continuous in it follows that a {x, t) is zero whatever x 

 and t may be. Therefore {x, t) = (x, t) for all values of x and t. 



59. Consider the improperly continuous theory. I proceed to prove the 

 following theorem : 



For a given initial characteristic function which possesses an associate, the 

 problem can admit of only one Solution C(§, 0 = k (|, 0 which satisfies, — of 

 course, in addition to (A^), (BJ, (Cg), and (Z)J, — the following conditions : 



i. k(|, 0 is a continuous function of |, and its single continuous associate, 

 viz., the limital function \i{x,t), is continuous in t and possesses a difFerential 

 coefficient m' {x, t). 



ii. There exists an every where dense aggregate (j^ =■ j^^j which is the 

 same for all possible Solutions k and which is, further, such that, for any 

 arbitrarily small but fixed quantity (?, it is always possible to find a positive 

 quantity Tv, different from zero and dependent on v, k and ö, for which 



