14 GANESHPRASAD, 



the case when 



D (X, X') x"°-' COS I «/^ {X') I , 



where ip (x') >- 1 ; the investigation naturally divides itself under three heads : 

 1. ^(a;')-<?(^), 2. tix')r^l[^), 3. t/.K)=-^(^)• 



14. It foUows from (I) that t^V {x, t) r*j f x'^'^^' cos U{x') \ e dx'. 



'o 



Let /, /i , stand for / , / , / x cos | (a;') } e dx' re- 



spectively, C being an arbitrarily large but fixed quantity. 

 Consider I^. 

 Put x' = 2\JTtj. 

 Then 



I, = i2\/rf + \f%'+\oB \^l.{2^ty)\e-y\ly. 



0 



Let ^{2\[ty) = i{;{2\/Ts) — i; = u — v, s being an arbitrarily small but fixed 

 quantity 



= f +J • 



0 S 0 



And 



J — cos mJ y cos ve ^ dy + sm m_/ y sm e a«/. 



Now 



^{2\[ty)-^{2^ta) = 2{y - e)sjt^' {2^i y), 



where 



Since t/> (a;') -< l ,^ it foUows that — {x') -< ^ • Therefore 



-Sit Ii:' (2 \/¥y)-<l 



and, for every value of y in the interval {s, C), v approaches zero with t. 

 Therefore 



l_|-7,: ?;2 /'^ l4-k . 



y cos ve dy and J y sin v e ^ f??/ 

 s s 



y e dy and 0, respectively, as t approaches zero. 



6 



1) This artifice was suggested to me by the procedure in Art. 4 ofDu Bois Reymond's 

 Memoir. 



