CONSTITUTION OF MATTER AND ANALTTICAL THEOEIES OP HEAT. 45 



Again 



{-|,F(-^,0| - \4t V(^,t)\^ - lim ~ = lim ^ = -c«. 



(Ot J<-o ( Ot )/— 0 « = + 0 Ol t — + 0 \jTtt 



Therefore (A) is meaningless when x — 0, jc or — n, 



(iii) Let T{x,0) = f{x) = f f^{x)dx, 0 < x<7t , f^{x) being the /'(ic) in 



" 0 



(ii) of Art. 24. Then f{x) is finite and continuous. At any irrational point 

 v . 



or rational point, — — p , with odd denominator, f^i^) = f (x) and is finite and 



V 



continuous; further, at any rational point, ^ — —r-, fl{x) exists and is finite. 



v 



At any rational point, , with even denominator 



T) H— CO -\ . 



f'(x + 0)-f'ix-0) ^ -~, D Standing for 2 ^^n + iy 



Now, proceeding as in Art. 20, it is easily seen that, as t approaches zero, 

 V(x, t) behaves as 



-fM{x, x') &"{x') dx', 0 < ^ < Ä , 



0 



1 f 0 



Now 



i. e., Sis - ^ r M'{x, x' + 0) &' {x') dx'. 



0 



Therefore 



= — oo. 



J « F(f , ()l = lim ± F(f , A = lim 



Therefore (A) is meaningless , when x is any rational point, 2^ , not only 

 because fix) is non-existent but also because F(a;, ^)| is infinite. 



(iv) Let T{x,0) = f{x) = f dxj cos ^l(^)^dx. Then f(x) as well as f (x) 



