12 ■ CtANESH pkasad, 



are also uniformly convergent in x' in the interval (0, %). Again fix') is finite 

 and inteecrable. 



Therefore 



Similarly, 



Therefore 



— — 2 / sin m {x + x') e f{x') dx' 

 1 'o 



/ f (x') 1 2 sin m (x + x')e j dx' 



^Tfix') @'(x + x')dx'. 



— 2 / "~ ^) ß / (^') f^^' 



?r 1 'o 



= -^ff{x')®'{x'-x)dx'. 



27t 'o 



^ ff{x')&'(x'-x)dx'. 



27C 0 



12. It is well-known that 



— (y—2n7cy — (y-\-2mty y'^ 



Therefore , when differs from 0 and 2;r by quantities which are numerically 

 greater than a positive quantity d, \&'{y, as ^ approaches zero. 



Therefore 



2n 



V (x, 0 = - — f^^fix') &' (x' - x) dx' + r (t), 



where e is an arbitrarily small but fixed quantity and 



l) Lc^M Stands for L-^MN where N is finita and positive (different from zero) but not 

 necessarily monoton. 



