18 



where 



GANESH PRASAD, 



U cos y — J y^'^^ cos |z ^ ^ ^Ui 

 Rsmy = _/ i/'^^ sin |? ^Ml- 



J(i/t^) 



Consider = j _ cos ridri where P(,r') and are equal to 



"z(i/Cx2\/r) 



~x''^'^^e and ^^^j respectively. 



The value of x' for which —P(x') attains its maximum is 



x' = d = y2{2 + k)t. 



Therefore, when C is sufficiently large, —P{x') constantly decreases as x' in- 

 creases from Cx2\Jt to 6. 



Patting l{\IC-x2 = let 



{m + ^)7t > ri^ > {m - ^) jt, 

 (m — r + i) TT > ? (1 Iß) > (m — r — \)Tt. 



Then 



^2 — I Q (v) cos Tjdr] 



'lim 



= ./ +./ +J +•••+./ 



(ni—^)7C (m — 1)51 (m— |jr) ^l/ö) 



= (- 1)» X 2 U„ g (^J - (3 (^) + ö (^-) _ . . . + (- 1)- Q (^_.) + (_ Jc^ Q J I 



where %, rj^, etc. lie within the limits of the first, second, etc. integrals respec- 

 tively, and 0 < (Zfo, \) < 1. 



Since the ^'s form a decreasing series it is easily seen that 



where 0 < Z; < 1. Therefore 



\I,\<4Q{r)o)<4Qi%) i-e. -4P(C>c2Vr). 



Thus 



\I,\<4:(Cx2\Jt f^^e~^\ 



Therefore, by choosing C snfficiently large, 7,, be made as small as we 

 please, in comparison with /j, provided that /, is not zero. 



Therefore, being the value of y when C is made infinite in (1), 



