CONSTITUTION OF MATTER AND ANALTTICAL THEORIES OF HE AT. 



61 



Für instance, if l = lO"' and q = 10'° , 



~ I T{x,0) I > 2, 



when 16 < w < 31. 



Uniqueness of the Solution. 



57, Consider first the continuous theory. I proceed to prove the following 

 theorem : 



For a given initial temperature, the problem can admit of only one Solution 

 V {x, t) which satisfies, — of course, in adäition to (A), (B), and (C), — the 

 following conditions : 



1. V (x, t) is continuous in t as well as in x and possesses a differential 

 coefficient v' (x, t). 



ii. There exists an everywhere dense aggregate g = |^,;| which is the 

 same for all possible solutions v and which is , further , such that , for any 

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

 quantity t^, diflFerent from zero and dependent on v, v and <?, for which 



I v' {x, t^, + r) — v' {x, ty) \ < 6, t < tv , 



whatever x may be. 



iii. There exists an everywhere dense aggregate G = { ^ | which is the 

 saw?e for all possible Solutions v and which is, further, such that||t;'(^;0L=:ll<2' 

 whatever | and t may be, p being a finite constant dependent on v. 



It should be noted that v {x, t) is defined only for ^ > 0. 

 58. If possible , let there be two solutions , {x, t) and (x, t), of the 

 problem. Then their difference 



is also a Solution of the problem, the initial temperature being zero. Therefore, 

 it follows from (A) that 



Now, going back to (ii) of the last article, let and t," be the values of 

 for and v^, respectively. Then it is evident that 



T,, Standing for the lesser of the two quantities t'^, tl' ; and, applying (2) to (1), 



{x, t) — (x, t) = to {x, t) 



(1) 



£o' {x, ty + r) — co' (x, ty) I < 2(3, T < T, 



(2) 



