64 



GANESH PRASA D, 



1 ü' K + - i^v) I < ^ "^v, 



whatever a; may be ; also, ti' (x, Q is a continuous function of x. 



iii. There exists an everywliere dense aggregate — } 1 1 which is the 

 Sflwee for all possible Solutions k and wMch is, further, suchthat \ 'i^v' {x,t)^^_~^\<zp^ 



whatever g and t may be, 2\ being a finite constant dependent on Ii. 



It should be noted that k(i, 0 and, consequently, ii{x,t) are defined only 

 for t > 0. 



If possible, let there be two Solutions k^d, 0 and \i^{i„t)- and let \)^{x,t) 

 and tjj {x, t) be the corresponding liinital functions. Then, expressing (AJ, (Cj), 

 and (Dj) in terms of u (.r, t) and in the notation of continuous analysis , it is 

 easily seen , by a procedure similar to that of the last article , that (x, t) 

 = v^{x, f) for all values of x and i. Therefore ki(|, = kj (|, t) for all values 

 of I and t. 



In the discontinnous theory , there can be no question as to the uniqueness 

 of the Solution because the Solution is non-existent and, in faet, impossible. 



