CONSTITUTION OF MATTER AND ANALYTICAL THEOEIES OF HEAT. 35 



. 2 



Therefore fa ! 



where b stands for the greatest value of \b^\ whatever m may be, being 

 J f"(a) cos mada. 



n 0 



Thus 



ii,<{^irwi+6)|^, i.e., (3) 



1 



hence f^\Y' (x, t')\dt' ^ - \ e-^' (^+^) - ."^^^ | 



and, consequently, P^ioc, t, ^ + tr) < 2 

 wliatever x, t, t+v may be. 



Therefore /I. < i Kl < {1 1 f » 1 + i,} i i, < | { i |f (.) i + i,| . (4) 



r(a;, ^ + T)-r(a;,^) = V(x,t + T)-V{x, i) = ^a^cosmx \e~''''^^^^^-e~'^'H^ 

 l 



hence Sl(t, ^ + t) < f] 



whatever t, t + r may be. 



Therefore T^< S |«™| < Ifi^f)! + &| • (5) 



Y' {X, ^) = — 2 * 

 1 



— / 1) e +2 sin?i«xe ; 



hence | r'(a;, ^)| < irNI + jM 



whatever x, t, t + r may be. 



Now ü( evidently less than or equal to the product of and the 



greatest value of | Y' (x^ t) \ within the interval {x^^ x^. Therefore 



^<A,{r(:^)i+|i^|. (6) 



Therefore it follows from the inequalities (3), (4), (5), and (6) that 



G,<\K^\f{n)\^- &| i {Tt^ + 16) K + ^ä], (7) 



5* 



