Colloquium publications.

74

THE MADISON COLLOQUIUM.

covariant. In particular, if we take (y) = (x), (z) = (.x2), we
obtain a covariant of F in the narrow sense used in these lectures.
In particular,
al a2 a3 al a2 as
(29) K= x x2 x, M = x x2 x3
X12 x2 x32 x14 x24 x34
are formal covariants of F. While the discriminant A, given by
(15), is a formal invariant, (16) is not. But
(30) A + A + 1 - a (mod 2),
(31) a = 2aibi -+ 2a 2 + ala2 + ala3 + a2a3,
a being a formal invariant of F. By (23), the B's are contragredient to the x's and hence to the a's, so that
(32) Ai = laiBi = -abi 2 + Zaiaj2 + ala2a3
is a formal invariant. For integral values of ai, bi,
(33) A, -A a 2ai(O + 1) (mod 2).
Any form with undetermined integral coefficients c1, 02, *.*,
taken modulo 2, has, by (21) of Lecture I, the invariant
(cl + 1) (c2 + 1).. Thus (16) is an invariant of (7) and hence
of F. Likewise from (19) and F itself, we obtain the invariants
(34) J = /S1/23, AJ = AH(bi + 1).
In (6) we made use geometrically of
(35) X = ulZi + U22 + U3x3.
Now F + tX2 is congruent modulo 2 to the quadratic form derived
from F by replacing each bi by bi + tu?2. Making this replacement in A, we see that the coefficient of t is congruent to K2, where
(36) K = a lu +- a2u2 + a3u3
is therefore a formal invariant* of F and X. Making the same
* Since (36) is a contravariant of F, ai(aC/dxi) is a covariant of F if C is.
Taking Q2, Q1, L as C, we get K, M, A, respectively.