CONSTITUTION OF MATTER AND ANALYTICAL THEORIES OF HEAT. 53 



50. As in the preceding parts, let V{x, t) stand for 



being given by 



1 ^ yyi^ f, 



"cT '''o + S '^m COS JWiC e , 



a„ = — / f [x') cos mx'dx'. 



7t 'o 



Now, since /'(i*^) is an associate to qp (|), it follows that = Therefore 

 F(a;, t) is an associate to % ^) ; and, since V (x, t) exists, it follows that 



X'{l,t) = [V\x,t)\^_.^. 



Therefore (Aj), (B^), (CJ and {D^ may be expressed in terms of V{x, f) 

 and in the notation of continuous analysis. Thns (A^) becomes 



f \ rix', t) - fix') I dx' = ./■'} v'ix, t')\ dt' ■ ■ 



— n 0 ■ ^ 



and, consequently, it follows from Art. 7 that (A^) is satisfied if for any value 

 of t', however small, \ [V i^, t')\ ^ _ ^\ is less than a finite quantity 



When 0<^<;r-l, 



Tix,f) = -^J Ci^',t)dl' = ±J x(r,04' = irf Vix',t)dx' 



(i^ ^ «™ sin ml — i 



= + 2] — cos mx i e 



2 1 l 



Now, since a. becomes indefinitely small with this series is uniformly 



convergent in t in the interval (0, + oo). Therefore Tix, t) is continuous in t 

 when 0<a;<jr — 1. Similarly it is proved that Tix,t) is continuous in ^ when 

 — l < < JE. Therefore (BJ is satisfied whatever qo (|) may be. 

 (Cj) is equivalent to the following: 



Lim F(|,if) = (fil) if the limit exists, or, the limit does not exist, and then qp^) 



is contained in the aggregate of values assumed by F(|, t) as t approaches zero. 

 It is easily seen that, if the condition relating to \\V' ix, t)\ ^^^\ be satisfied, 



1 , , > , , a„sinw| , — m^i 



-^Qilt) =fc'iit')dt' =/ \v'ix,t')\^_^dt' = 2- -1), 



— 7t < I < 3t. 



Now this series is uniformly convergent in |. 



