Colloquium publications.

76

THE MADISON COLLOQUIUM.

are contragredient with a1, a2, a3 and hence with x1, X2, X3, and
therefore cogredient with ul, U2, u3. Thus (39) yields the formal
covariant
(41') W' = S(B1B2 + B12 + B22) 32 + -B124223.
From this or (39'), we obtain the modular covariant
(41) W = 12(/22 + 1)%32 + 2(01 + 1)%223.
In these notations (29) become
(42) K = Bai, = = alil(z22 + X2X3 + X32).
Finally, by (16) of Lecture III, we have the universal covariants
X1 X2 X3
(43) L3' = X12 X22 x32 12X2232,
X X21 X4 X4 J34X1 2 1X2
The covariant line L- 0 of a non-degenerate conic F 0 is
determined by the three (collinear) diagonal points of the complete
quadrangle having as its vertices the apex (a) and the three
intersections of F = 0 with its covariant cubic curve K 0.
FUNDAMENTAL SYSTEM OF COVARIANTS OF THE TERNARY FORM F,
~~ 9-32
9. Invariants of F.-A fundamental system of invariants of F
is given by A, A, J. It suffices to prove that they completely
characterize the classes of forms F under the group of all ternary
linear transformations with integral coefficients modulo 2.
This is evident from the following table
Class A A J L
1X2 + X32 1 0 0 X1 + 2 + X3
X1x2 + x12 + 22 0 0 1 0
X1X2 0 0 0 x + X2
X12 0 xi
12 0 1 0 1
0 0 1 1 0