44 ganeshpräsad, 



6^ being a positive fraction dependent on x, t, r. But T ' (x, t) is continuous at 

 {x, t). Therefore (C) follows from (C). 



42. Consider the Solutions of Fourier's type, viz., 



T{x,t) = V{x,t), 

 T{x,())= fix), 

 d 



When t > 0, it is evident that ^ V{x, t) and V" (x, t) are continuous at 



[x, t). The important case is that of ^ = 0 ; and it should be noted that , in 

 Order that (A), (B), (C) have any meaning, it is necessary and sufficient that 



i— F(.r, ^5)1 and f"{x) exist and be finite. 

 \dt J< = o 



Now, proceeding as in Art. 20, it is easily seen that, if / " {x) be continuous 

 and f ' (ä), f (— 7t) be zero , 



lim 4t ^(^,0 = lim V"{x,t) = f" (x). 



t = + 0 i! = + 0 



But, since lim y{x,i) exists, it must equal l-^^C^, Of • Therefo 



t = + o ot [Ot ]i = o 



d 



— - V{x,t) and V"{x,t) are continuous at (x,t) whether t>ov = 0. 

 dt 



Therefore , in order that (A) , (B) and (C) have meaning and necessarily 

 follow from (A) , (B) and (C), it is necessary that f"{x) exist and he finite; 

 and it is sufficient that f"{x) be finite and continuous, and f {n), f {— jt) 

 he zero. 



Illustrative Examples. 



43. The foUowing simple examples suffice to illustrate the salient features 

 of the criticism of Fourier's theory: 



(i) Let T{x,0) = fix) = a;^7^>l. Then fix) as well as fix) and fix) 



d d 



are finite and continuous. But, as t approaches zero, F(ä, ^) and — F(— tt, ^ 



2 ' d 



behave as — . Therefore ] — Ffjr, 0 i and i^~r F(— jr, are infinite, 



Sjjtt [dt ^ li = o [dt '\t=o 



and, consequently, (A) is meaningless when x = jt or — jt. 



(ii) Let T(x,0) = f{x) = x, 0 < x ^ n. Then f"{x) does not exist at 

 X = Q; also, 



re 



