58 Dr. I. J. Schwatt : JSote on 



The problem is therefore reduced to the finding of 



J iU»-^-D»— (a?"+l) [ J 



The greatest value of ft is c, and that of c is p — 1, c being 

 by hypothesis less than p. We must therefore consider the 

 following two cases. 



(i.) /3=c=p-l, then (8) becomes i J^, . . (9) 

 and (ii.) /3<p-l, then (8j changes to j ^ m ^ + ^ , . (10) 



In either case let ?v denote one of the n nth roots of — 1 , 

 then 



^ _ $ K _ A. ,±Xx) 

 n . -i — ^ ~~ ~~ — r -7-7 \ ? • • \ ±A -J 



X n +1 K =\X — T K X — T K <p K {x) 



where "fi 1 / . " / . 



fW= n (a— r fl ) n (z — rp). 



Multiplying both sides of (11) by x — r K , and letting #=r K 

 we obtain 



x-r K l 1 "1 1 

 A K = A' a — — - = #*-—— r = 



Therefore 



a+l 



,a 1 » ™ a+l 



=— ±2.— — (12) 



Now ?«+i ^ 



where 

 *=0, -1; 1,-2; 2,-3;....; ^,-^iTL 1 ; '1=1; if n is odd, 1 



and 

 « = 0, —1 ; 1, —2 ; 2, —3 ;....; ■ — „— , — „ ; if n is even. 



