10 GANBSH PRASAD, 



Now, 



t t t 



f V'{x,t')dt' = f+fv'ix, t')dt' 

 where #o > 0 arbitrarily small and fixed. 



V {x, t') = — 2 ""'m '"^i^ ^''^ ß * . 

 1 



and, since this series is uniformly convergent in t' for t'>0, it fis integrable 

 term by term in the interval (^g, 



Therefore the right side of (A') is equal to 



2 — sin mx e"'""' * - |] sin mx e~™' + T^V' (ic, t') dt'. 

 But it is well known that a, becomes indefinitely small with — • Therefore 



y,—^sminxe = ~ smmx+ 6,(t,,) 



where ß^itg) becomes indefinitely small with f^. Thus, if for any value of t', 

 however small, \V' (x, t')\ is less than a finite quantity P{x), the right side of 

 (A') is equal to 



y. — ^ sm mx e — >] sm m.x — 6, (tA + m.F (x) 

 im X m 1 \ ü/ u \ / 



where 0 < | G | < 1. (2) 



Therefore , by choosing sufficiently small , the dilference between (1) and 

 (2) can be made as small as we please. Therefore (A') is satisfied, 



8. When \V' {x, t')\ does not remain less than a finite quantity F{x), as t' 



J 



approaches zero, then / V'(x, t')dt' is meaningless and, consequently, Q{x, t), the 

 'o 



quantity of heat which flows across unit area, at x, in the interval (0, t), is in- 

 determinate. I will, therefore, Supplement the hypothesis (a) by the following 

 hypothesis : 



If the values of x, for which P(x) exists and, consequently, according to 

 (ß), Q(x, t) becomes indefinitely small with t, form an aggregate which is every- 

 where dense within the domain of x, i. e. the continuum (— n, then Q, (a;, t) 



becomes indefinitely small with t for every value of x within the domain. [y] 



9. Consider the condition (C). It foUows from the two preceding articles 

 that, if the values of x^ for which P exists, form an aggregate every where 

 dense within the interval {—tc, ti), 



1 „ , ^ a„ . —mH " a,„ . 



— ^^^("C, üS) = 2u^^^mmxe —y,^^^mmx, — n <z x it. 

 iL ^ 1 m -Y in 



