60 GANESHPRASAD, 



(viii) 6'(|, 0) = I" — ^ gives a better approximation to the impossible 



ö 



initial state, T^{x) = x', thaa C(§, 0) = |\ For, the temperature 0), de- 

 fined by C'd, 0) = |^ — ^ , equals as long as x is in the interval (0, tc — l). 



o 



Similar approximations to the impossible initial states and 

 are given by the admissible characteristic functions 



and 



C'(l,0) 



= r 





- r 





= l 







Ci^, 0) 









fc3 ja 



49 31 

 respectively. 



(ix) Consider the impossible initial state T.(x) = xsinqx, | sin | > 0. 

 However large q may be, the temperature T{x,(y) defined by the admissible 

 characteristic function, 



equals x sin qx as long as x lies in the interval (0, :r — l). When 7t — l-<:x<7t, 

 -^1 T(x, 0) -X sin qx\< (^q + <l^q+ lO''. 

 Now, if q be integral, putting x — tk — rj + l, 1 < < 21, 



T(x, 0) = ^ siii'gl^^ I (1 ~ cos (j"»?) + rj cos gi^ ~ l cot gl sin g'r; | . 

 Therefore, when qrj = {2u + 1) n being integral, 

 ^ ■ Tix,0) ' ^ 



' ^ ' I jp ] sin 5l| 



Thus , if 0 < I sin gl I < , say , , T(x, 0) ceases to be an approximation , to 

 T, {x), ever so crude, at any point in the interval - l, 7t), where 



, 2n + 1 



X = 7t + 1 7t. 



