6 



GANESH PEASAD, 



Given the initial characteristic function C 0), find the characteristic func- 

 tion at any subsequent time. 



We then go on to obtain the analytical representations of the conditions 

 of the phenomenon; these are the conditions {A^, (B^), (CJ and (DJ of Art. 47. 

 In Art. 48 attention is drawn to the definition of temperature , as given by 

 (^2) , which indicates the Classification of initial temperatures as possible or im- 

 possible. In Arts. 49 — 53 we discuss characteristic functions of Fourier's type. 

 We begin by expressing (Ä^), {B^, (CJ and (Z>J in terms of V{x, t) and in the 

 notation of continuous analysis ; we thus obtain in Art. 51 the group of con- 

 ditions which is necessary and sufficient in order that C{i„t) = %{i„t), (7(^,0) 

 being (p(l): here t) = F(|, 0 and (p® = f{l). 



These conditions are as follows: 



i. For every value of | , there exists a finite constant P such that , for 



any value of t, however small, j-^F(Ä;, 



is less than P, — ;r < ^ < 



ii. Lim F(|, t) = (p (|) if the limit exists ; or, the limit does not exist, and 



t = -\-0 



then is contained in the aggregate of values assumed by F(^, t) as t appro- 

 aches zero. 



Making use of the results of Arts. 19 and 21 we find in Art. 52 necessary 

 and sufficient conditions, for qc^), of an extensive applicability. These conditions 

 at once indicate the Classification of initial states as stable or unstable, oscilla- 

 tory or non- oscillatory , admissible or inadmissible. This Classification is dis- 

 cussed in Art. 53. In Arts. 54 and 55 we discuss approximations to impossible 

 initial temperatures. We conclude the exposition of the theory with Art. 56 in 

 which nine examples are given to illustrate the salient features of the theory. 



This part ends with Arts. 57 — 59 in which we give a careful discussion of 

 the question of „the uniqueness of the Solution", specially bearing in mind the 

 fact that there may be Solutions other than functions of Fourier's type. 



(6) 



In Part IV. we discuss briefly the theories, expounded in the previous parts, 

 in so far as they illustrate the nature of the relation of mathematical analysis 

 to physics. 



