2 GANESHPRASAD, 



tensive class of Solutions in terms of functions of the type 



V(x, t) — ^ «0 + S mxe t> 0, 



wliere a,„ = — / f (x') cos mx' dx' , f (x) being any integrable function; such func- 

 tions V{x, i) we call, for the sake of convenience, functions of Fourier''s type. 



(3) 



In Part I. we first find the analytical representations of the conditions of 

 the phenoroenon. These are the conditions (Ä) , (B) and (C) of Art. 4. In 

 Art. 5 attention is drawn to the fact that there may be Solutions of the pro- 

 blem other than functions of Fourier's type. Throughout the remainder of this 

 part we discuss Solutions of Fourier's type. We begin by expressing (A) , (B) 

 and (C) in terms of V(x, t), f(x) being the initial temperature. We thus obtain 

 in Art. 10 the group of conditions which is necessary and sufficient in order 

 that V{x, t) be the Solution of the problem. These conditions are the following: 



i. For every value of x ot , at least, for an everywhere dense aggregate 

 of its values , there exists a finite constant P such that , for any value of t, 



however small, 



< P. —7t<X<.7C. 



ii. Lim V{x, t) =f(x) if the limit exists ; or, the limit does not exist, and 

 t—+o 



tben f{x) is contained in the aggregate of values assumed by V(x, t) as t appro- 

 aches zero. 



In Order to find the nature of the restrictions which these conditions impose 

 on f{x), we carefully investigate in Arts, 11 — 21 the behaviour of V'{x, t) and 

 'V{x, t) when t approaches zero. The final results of this investigation, in which 

 we make use of Du Bois Reymond's Infinital Calculus (Infinitärcalcül) , are 

 given in Arts. 19 and 21 Making use of these results , we give in Art. 22 



1) It would be convenient to explain tlie notation which we use: If A{y) and B(y) be two 

 functions of y such that is positive and monoton in the neighbourhood of 2/0 ; then it is 



Said that Ä{y)^^B{y), as y approaches y,,, according as has a limit which is infinite, finite 



or zero. This is Du Bois Reymond's notation. We have found it convenient to introduce four 

 new Symbols, viz. , >*-, n»j, -•< and ±>-. By Liy) Bliy) , we mean L{y) ^ 3I(y)N(y) 

 where N{y) is finite and positive (diiferent from zero) but not necessarily monoton. For example, 

 — >•- Z f— 1 (2 -f- cos — ^ as y approaches zero. If, with the approach of y to y^, G{y) makes an 



r \yl\ yi 



indefinitely large number of oscillations, from positive to negative values, with indefinitely increasing 

 amplitude; then we say that C(y)±'>~ 1 as 2/ approaches yo- For example, icos^^^±>-l as 

 y approaches zero. 



