CONSTITUTION OF MATTER AND ANALYTICAL THEORIES OF HEAT. 43 



I proceed to investigate what further conditions T(x, t) must satisfy in 

 Order that (A) and (C) follow from (A) and (C). 

 41. It follows from (A) that 



X -\- X-^ t -\- t 



f {T(x',t + T)- T(x',t)\dx' = f \T' (x + x„t')-T'{x-x„t')\dt', (1) 



X — t 



— 7C<X + X^-<.7t, 



— 7t X — X, 7C , 

 t>0, T > 0. 



Now the first integrand in (1) is continuous in x', and the second integrand 

 is continuous in t' ; therefore (1) becomes 



• T{x + (ix^, t + r)- T{x + (ix^,t) T'(x + x„ t+ O.r) - T' (x - x„ t + B^r) 



T 2x^ 



i. e., 



where the 6's are functions of x,x^,t,r such that 0<(|6|, |03|)<l,O<;(0j,G.J<l. 

 d 



Therefore, if — T{x,t) and T{x,f) be continuous at {x,i^, (1) becomes 



where Z can be made as small as we please by choosing x^^ r sufficiently small. 



-TT- T{x, t) and ^ 

 dt ^ ^ ' dx^ 



d 



Therefore (A) follows from (A) if ~ T{x,t) and ~^T(x,t) be continuous at 



{x,t). 



It follows from [C) that 



lim / T'{x,t')dt' = 0, ^>0,r>0, 

 x=n — 0 



and 



t + t 



lim / T' (x, t') dt' = 0, r> 0, r > 0. (5) 



x ~ — n-\-0 ' t 



Now T' {x,t') is continuous in t'. Therefore (5) becomes 



and 



lim T'(x,t + b^T) = 0 



X -- Tt 0 



lim T'{x,t + e,T) = 0 , , (6) 

 X = — 7r-\- 0 



