128 CHAPTER V. 



IS. V*i**th "iJiJ^te 



li^~ ttij^ctijs = (j' 3 = 1, . . ., w&; j&^FJv Jz)- 



If neither r nor r 1 is divisible by p, we may drop the numerical 

 factors from these m equations. 1 ) But 



being the determinant of $. Hence we have 



Hence only one element of each row of the matrix for S is not 

 zero. The determinant of S being not zero, the non- vanishing 

 coefficients lie in different columns as well as in different rows. 

 Hence S merely permutes the terms of the sum O r . 

 Suppose next that r 1 is divisible by p and set 



r 1 gp' (s :> 1), 



where g is not divisible by p. We now consider the case g > 1. 

 We make use of the following equations of the set 116): 



1=1 





 U; 



( ft (Y\ S , Irn III ^t * X N. 



(^^'-t- 1 ;- ^ /^ a (. 9 _ i)i* a jp* ) a =o (;=!.. m- j 4=; 9) 



=i 



of which the first two alone occur when m = 2. We may verify 

 that the numerical factors are not divisible by p. 2 ) Then, since | a,-y 



It follows as before that S at most permutes the terms of <t> r . 



1) If m = 2, only the first two equations occur. The same conclusion 

 follows in this case that was derived for m ^> 2. 



2) This result follows by inspection from a general theorem on the residue 

 of a multinomial coefficient taken modulo p given in the author's Dissertation, 

 Annals of Mathematics, 1897, 14, p. 75. 



