228 HISTORY OF THE THBOEY OF NUMBERS. [CHAP, vni which for g(x) = 1 yields the first formula of Lerch. Next, if the k'a are primes and q is a prime distinct from them, **-! * 2 £(sn-g; A?!,..., *,; x) = S (fc,-l, n; g)0(*j). x-O 1-1 Finally, he treated f(x) of degree d=fcj—2, whose constant term is prune to each ki and coefficient of #d~* is divisible by the prime &M if t< &«—&„. Gegenbauer30 noted that, if p—1 — ju is the rank of the system (3) modulo p, the congruence, satisfied by the distinct roots 5^0 of (2) and by these only, is given symbolically by JL, * Y | ai+k I =0 (mod p) ft *-0,..., p-2). dOi oa0/ He obtained easily Kronecker's25 form of the last congruence. He gave necessary and sufficient conditions, expressed in terms of a complicated determinant and its /u—1 successive derivatives with respect to ap_2, hi order that (2) and a second congruence of degree p—2 shall have /* common roots 7^0, and found the congruence satisfied by these ju common roots. He deduced determinantal expressions for the sum <rr of the rth powers of the roots of (2), and for the coefficients hi terms of the <r's. Michael Demeczky31 would employ Euclid's process to find the g. c. d. G(x) modulo p of (2) and xp—x. If G(x)=Q (mod p) is of degree v it has v real roots and these give all the real roots of (2). Multiple roots are then treated. The case of any composite modulus is known to reduce to the case of pwt p & prime. If (2) has X distinct real roots, not multiple roots, we can derive X real roots of/(#)=0 (mod pT). If pi,..., pn are distinct primes and if f(x)=Q (mod pt) has X; real roots, then f(x)=0 (mod p:.. .pn) has Xi.. .Xn real roots, and is satisfied by every integer x if the former are. Various sets of necessary and sufficient conditions are found that/(o?)sO (mod ?n=npir<) shall have m distinct real roots; one set is that/(z)==0 (mod p *«) identically for each i. L. Gegenbauer32 proved that a congruence modulo p, a prime, of degree p—2 in each of n variables has a set of solutions each ^0 if and only if p divides the determinant of a cyclic matrix A* A1 ... A' A''1 A° ... Ar A1 A2 ... A° where A" is itself a cyclic matrix in B°,..., Bri~1; etc., until we reach matrices in the coefficients of the congruence. An upper limit is found for s°Sitzungsber. Ak. Wiss. Wien (Math.), 98, Ila, 1889, 652-72. "Math. u. Naturw. Berichte aus Ungarn, 8, 1889-90, 50-59. Math. 6s Term&i Ertesitfi. 7, 1889, 131-8. 32Sitzungsber. Ak. Wiss. Wien (Math.), 99, Ila, 1890, 799-813.