16 GANESH PRASAD, 



Now, putting —P(x') = QiTj), 



= +/ +/ +•• +/ 



/a = / Q in) COS 7] dr] 



71 37t 2r— 1 



2r+l _ , , 2r-l 

 where r]^ ^ — ;r < (0) < ^ — n, r bemg integral. 



/•i/'(Cx2v^) 



Suppose t = so that ri^ = mit, where m is integral. Then | 



J 7t 



7t ^^~'9 



is niimerically greater than (— 1)"' 1 and therefore still greater than 



and it has the sign (—1)". 



7t 



j IS numerically less than 7cQ\rj^—-^j and has the sign (—1)"'"^'. 



37t 



37t 5« 2»-— 1 



f +/ +■■•+/ 



= (_ir><2|(2(^,)-4K'^3)+---+(-irö(i-.)+(-ir/^«('?:)l 



where ij,, etc. lie within the intervals (vi-^, ^'~^)' (^'~"^' ^»""x)' 

 etc. respectively, and 0 < 7^,. < 1. 



Therefore Qi%), Qirj^), etc. form a decreasing series; and, consequently, Zj 

 has the sign (— 1)" and is numerically greater than 



||- cos ^ (2 [% [v. - f) 



provided that this expression is positive, 

 Now, let 



^ix) = Vi — Q-, 



TT 



