Colloquium publications.

INVARIANTS AND NUMBER THEORY.

69

of the linear covariant AmZbixi. We shall see however that there
exists a more fundamental linear covariant.
4. Covariant Line of a Conic.-Since we shall later treat in
detail the case m = 3, we shall replace (1) by the simpler notation
(14) F(x) = alx23 + a2x1X3 + a3X1x2 + blx12 + b2x22 + b3X32.
Its apex is (al, a2, a3). Its discriminant (12) is
(15) A = F(al, a2, as) ala2a3 + a12bl + a22b2 + a32b3.
The invariant (13) becomes
(16) A = alct2a3 (ao = a +1).
Consider a form (14) with integral coefficients and not the
square of a linear function. Then not every ai is congruent to
zero modulo 2. By an interchange of variables we may set
as3 1. Replace xl by X1 + a1x3 and x2 by X2 +- a2x3. We get
X1X2 + blX12 + b2X22 + Ax32.
Let A - 1. Replace x3 by X3 + blX1 + b2X2. We get
(17) < = X1X2 + X32.
The only real points on — 5 0 (mod 2) are (1, I, 1), (1, 0, 0),
(0, 1,0). In addition to these and the apex (0, 0, 1), the only
real points in the plane are (1, 1, 0), (0, 1, 1), (1, 0, 1). These
lie on the straight line
(18) X1 + X2 + X3- 0 (mod 2).
Hence with every non-degenerate conic modulo 2 is associated
covariantly a straight line.
The inverse of the transformation used above is
X1 = x1 + aix3, X = x2 + a2x3,
X3 = blXl + b2x2 + (1 + albl + a2b2)x3.
It must therefore replace 4 by the general form (14) having