60 CHAPTER V. 



Hence a necessary condition is that 



<!>_! = (<- 1,2, ...,*- 2). 



The condition 2 is evidently a necessary condition. 



Suppose, inversely, that 1 and 2 are satisfied. Consider the 

 equation satisfied hy the marks 



J7 h - <(ft 



./ = = 



the sum of the ?w th powers of whose roots is denoted by (? m . Then 



- 1 --P"- 1= - 1, 



since all hut one of the ^(fi/) are by 2 and hence have unity 

 for their (p n 1)*' powers. Applying Newton's identities 40) , we 

 readily find 



y .= [i = 1, 2, . . ..j? w - 2; z EEJEE (mod p)] 

 0p= 6z P = -'= <Spp= G p n= 0, y jB _ 1 = 1. 

 To determine y p , yajo; , we apply the identities 42), viz., 

 ^4- 71 <?*- 1 4- ^2 fffc - 2 H ----- h v Gk - P n = (A- ^ i> n )? 

 which here reduce to the form 



51) tffc-h y p 6 k -p 4- 7'2 P <>k-2p-\ ----- r y P n P 0k p n+ p + ^_ i (?A- - ^ w + 1 == 0, 

 since by 2, _ i 



Furthermore, since any mark equals its (j) w ) th power, we have 



Applying 51) for 7s =^ rt -h j> 1, we find 



y /> <y /) _i= 0. 

 More generally, for k = p n -\- Ip 1, Z ^p n ~ 1 1, we get 



^ J0 (? p _ 1 = 0. 

 Hence y^= 0. We have therefore the result 



so that the marks Ofe) form a permutation of the marks ^ of the 



