CONSTITUTION OF MATTER AND ANALYTICAL THEORIES OF HEAT. 23 

 n =ct> 



(i) Let T{x, 0) = f{x) = '^a" cos (b^x), where a is a positive constant less 

 than 1, b is an odd integer, and ab>l + ^- This is Weierstrass's function. 



Consider the aggregate ]~^[ where m and \M\ are positive integers and 



jjfl is prime to b and less than 6™. The aggregate is everywhere dense within 

 the intervaJ {—n, 7t). 



For any point = -^r~ of this aggregate, 



\D{x, x')\ = 



n = m — 1 



1 



^ a" sin (b" x) sin (5" x') | 



(ab)'" — ab 



x' 



ab — 1 



hence \D{x^, x')\rmjl, and consequently \V' (x^, t)\otül. 



Also f(x) is continuous. Therefore V{x, t) is the Solution of the problem 

 and the initial state, T{x, 0) = f{x), is stable and non-oscillatory. 



« = CD ('^J^Jq\ 



(ii) Let T(x, 0) = f{x) — 2 — —i s > 2, 0<x<7t, where {nx) represents 



zero, when nx is an integer or half an odd integer, and in all other cases, the 

 positive or negative difference between nx and the whole number nearest to it. 



V 



At any rational point 2p,-\-\ ^^^^ denominator, 



lim D {x, x') = ^ 



= +0 1 



W =00 



hence lim V {x, 0 = S ~^:tj 



and, consequently, V'{x,t)r»jl. 



Also f ix) is continuous at any irrational point or rational point x -r ; at 



any rational point, -g^, with even denominator, f(x) has a discontinuity of the 

 first kind but such that 



f{x) = 1 |/(^ + 0) + /'(^'-0)i = lim M(x, x'). 

 Therefore f{x) = lim V{x, i). 



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

 though discontinuous, is stable and non-oscillatory. 



(iii) Let T(x, 0) = f(x) ='2"-^^^^iTr^, s>l, O^x^jt, where 



1 



