Chap. VIII] NUMBER OF RoOTS OF CONGRUENCES. 229 
the number of sets of solutions each not divisible by p. He proved that 
s EzlL n 
S ttjXj ^ + S a,+jX,+j-{-h=0 (mod p) 
y=i i=i 
has p""^*"^ sets of solutions. Of these, 
have each x^^O, where r is the number of the 2' integers 
6 =1=01 ±02='= ...=*= a, 
which are divisible by p. The number of sets of solutions of 
s Pui n 
l^QjXj 2 +i:a,+jXs+j+b=0 (modp) 
i=i 3=1 
is expressed in terms of the functions used for quadratic congruences. 
*E. Snopek^^ gave a generalization of Konig's criterion for the solva- 
bility of a congruence modulo p. 
L. Gegenbauer^^ proved that if the p congruences 
S Zk^x^-^-'=0 (mod p) (X = 0, 1,. . ., p-1) 
A: = 
have in common at least p—p distinct roots not divisible by p then all 
p-rowed determinants in the matrix (^^^x) are divisible by p. The converse 
is proved when a certain condition holds. By specialization, Konig's 
theorem is obtained. 
Gegenbauer^^ proved that, if r is less than the prime p and ii Zq,. . ., 2r_i 
are incongruent and not divisible by p, the system of linear congruences 
(4) s' h+,y,^0 (mod p) (p = 0, 1,. . ., p-2) 
has all its sets of solutions of the form 
(5) 2/*^'sa,2,* (A: = 0, l,...,p-2) 
x=o 
or not, according as the matrix (bk+p), k = r, r+1, . . ., p — 2; p = 0, . . ., p — 2, 
has a p— r — 1 rowed determinant prime to p or not. Next, if 
(6) S 6;fcX^=0 (mod p) 
A:=0 
has exactly r distinct roots Zq, . . ., z^-i each not divisible by p, every sys- 
tem of solutions of (4) is given by (5), and conversely. By combining this 
theorem of Kronecker's with the former, we obtain Kronecker's form of 
Konig's theorem. 
"Prace Mat. Fiz., Warsaw, 4, 1893, 63-70 (in PoUsh). 
'^Sitzungsber. Ak. Wiss. Wien (Math.), 102, Ila, 1893, 549-64. 
"Monatshefte Math. Phys., 5, 1894, 230-2. Cf. Gegenbauer." 
