CONSTITUTION OF MÄTTEE AND ANALTTICAL THEORIES OF HEAT. 59 



(v) Suppose that the discriminating aggregate G of the slab is (tc, RJ, R^^ 



M 



Standing for the aggregate of all the rational numbers with an integral 



power of 49 as denominator, in the interval (0, n). 

 Let 



1 



£o„, having the same meaning as in (iii) of Art. 24 and G being taken to be iden- 

 tical with the sub-aggregate represented there by G^. 



The initial state C (|, 0) = (|) is admissible, stable , and non - oscillatory. 

 But, for any slab whose discriminating aggregate is {tc, i?J or (w, R^, 



1 '^r 



would be inadmissible. 



(vi) . Consider the function x^. There does not exist any even function C(^) 

 which can be connected with by the equations 



^1 x—i 



n — x+i 



x—i 



Therefore x' = Ti(x) cannot represent a temperature ; and, consequently, 

 an initial state in which the temperature is supposed to be x'^ must be an im- 

 possible one. The initial state defined by C(|, 0) = is an approximation to 

 T,ix); for, 



T(x,0) = ~l l^dl = ~l x'dx = x'+^, 0<x<jt-i, 



T(., 0) = 1 ,/' «Vi = 1 = ''' + '''^-'> + <"-')', 



jt — l < < ;e, 



and, consequently, 



ir(^',o)-^-| i 10-' 



whatever ;r may be. 



(vii) If n be not too large, an approximation to the impossible initial state, 

 T^{x) — X" , n>l, 0 <x <,7t , is furnished by the temperature T{x, 0) defined 

 by the admissible characteristic function C(^, 0) = and 



\T(x,0)-x''\ 2nl 2n 



TC" X O 



whatever x may be. 



