24 GANESHPRASAD, 



^^•(2/) = 0, «/ - 0; 



further, = i»»,. i, G-^ = {(Onx] are everywhere dense subaggregates of 

 Cr = I G}„ I , the aggregate of rational numbers , arranged according to Cantor, 

 such that (r = G^ + G^ and the rth and Ath terms of G^ and G^^ >?,th 

 and «^th terms of G, respectively. 

 At any point x = o,,^ of G^, 



|lim F'(:r, Ol = |lim a;')] = 



<=:+0 x' = +0 



r =00 1 



hence \V' {x, ^)| r«j 1. 



Also f{x) is continuous at any irrational point or any point belonging to 

 G^; at any point o,,^, of G^, f{x) has a discontinuity of the second kind and 



W{X — 09,, ) WiX — 03n) 



behaves as (x) -\ — , where f\ {x) — f{x) — and is continuous. 



Therefore at any irrational point or any point of G^ 



lim V{x, t) = f (x) ; 



at any point co,j^, of G^, V(x, t) makes, within any indefinitely small interval 

 (0, ^J, an infinite number of oscillations , about f{x)^ of finite amplitude not 



greater than • 



Therefore V{x, t) is the Solution of the problem and the initial state f (x) is 

 stable and oscillatory. 



(iv) Let T{x, 0) = f{x) = 2 , s>l, 0 < x < ;r, where 



1 »^-^ 



W,{y) = cos 1^(1-)}, \y\>0., 



^,{y) = 4, y = 0. 



V{x, t) is the same as the V{x, t) of (iii). Since *fi(0) >2, fi^n) is not contained 

 in the aggregate of values assumed by V(c3n^, t) as t approaches zero. Therefore 

 the initial state is unstable, and, so to speak, runs down instantaneously to the 

 initial state of (iii). 



(v) Let T(x,0) — f{x) where f{x) is equal to the f{x) of (ii) at all the 

 points in the interval {—n, tc) with the exception of an aggregate of points of 

 zero content. Then V{x, t) is the same as the V{x, t) of (ii) ; and the initial 

 state, being unstable, instantaneously goes into the initial state of (ii). 



