Colloquium publications.

86

THE MADISON COLLOQUIUM.

x's, of powers of the covariants (77) can be reduced by means of
the syzygies
JL = 0, AL2= AF, (A + A + J + 1)(FL + K) = 0,
(78) AK= 0, FL2 + (A + A) L4 + AF2 + Q2 = LK,
F3+ Q2F= L3K + (A + J)K2 + (A + 1)LG + (A + 1)Q1,
to a sum of covariants of order w given in ~~ 11-18 and a linear
function, with covariant coefficients, of K, Q1 and
G = Q2L + L5 = S2[I3(1 + 1)x32 + (0103 + 1)x3X1
(79) + /31(3 + 1)x12] + XIX2X3[(31 + 02 + 03 + 1)
X (X1X2 + X1X3 + X2X3) + I(Oi + 1)Xi2].
Here G and K, given by (42), are of rank 1, while Qi= -22+x2 ( )
is of rank 2. As this theorem is not presupposed in what follows,
its proof is omitted. However, it led naturally to the important
relations (75) and (79) and showed that no new combinations
of the covariants (77) of rank zero yield covariants of rank > 0,
a fact used as a guide in the investigation of the latter covariants.
REGULAR COVARIANTS Rrn0, ~~ 20-22
20. A separate treatment is necessary for covariants (76)
with n = 0. Then each fi is a function of the coefficients aj, bj.
Since the factor ~3m of the part f3~3m of Ro free of x3 is unaltered
by every linear transformation on xl and x2, f3 is a linear combination of the functions (49) and their products by b3. Also,
f3 must be unaltered by
(80) xi = x1' +: al' = al + aa, b3' = 3 + b + a2.
Both conditions are evidently satisfied by the ternary invariants
and by a3 and q, in (47). In view of (53), we may employ
AJ, J, a3A, A, a3J, qA, A
to replace in turn
b3CalC2a3q, b3ala2a3, asj, j, a3b3q, alo2a3q, ala2a3,