Colloquium publications.

INVARIANTS AND NUMBER THEORY.

81

combination of the covariants (65), while C' is a covariant whose
leader is an invariant. For o = 2,
C' = Sx32 + S1X3X1 + Sx12 + x20.
This is transformed by (51) into a function having S1 as the
coefficient of x12. Since S is an invariant, Si = S. Thus every
coefficient of C' equals S. Then (51) transforms C' into a
function in which the coefficient of x1'x2' is zero, so that S = 0.
Hence every quadratic covariant is a linear function of
(66) F, AF, AF, JF, L2, AL2.
14. There remains the more difficult case of covariants (60)
of order o = 4n + 2 > 2. If Si' is the function obtained from
Si by the substitution (50), then
(67) SI' = Si, S2' = S + S1 + S2.
Now S1 is unaltered also by the substitutions (22) and
(68) as' a3+ a2, b2' 3 b2+ b3 + al (mod 2),
induced on the coefficients of F by the transformations (8) and
(69) Xi = Xi', X2 = X2, X3 = X3 + X2.
15. A fundamental system of invariants of F, under the group r
generated by the transformations (8), (51) and (69), is given by
A, A,, a2, b, a3a aa2 and
(70) = bl(b3 + a2).
It suffices to prove that these seven functions, which are
evidently invariant under r, completely characterize the classes
of forms F under r. There are six cases.
(i) b3 — a2 = 1. Replacing x1 by x1 + al12 and x3 by
x3 + a3x2, we get
F = Xli2 + Ax22 + x32 + x1x3.
(ii) b3 1, a2 0, a2 a2 a 2 1. Replacing x3 by 3 + a3X1,
we get

F = Axl2 + b2x22 + X32 + X2X3.

7