Colloquium publications.

92

THE MADISON COLLOQUIUM.

tions of the proper degree in the x's, of the generators (77) of
covariants of rank zero, I found the syzygies needed to reduce
C1K + D1G to an expression differing from the above CK + DG
by a covariant of rank > 2, in which those of rank 2 are linear
combinations of K2, KG, G2, W, Q1 and the new one
V = GF2 + AQ2G+ (A +J q+ 1)Q2FK
(89)
where
where + AL3K2 ~ AL3Q1 = }22x33v + ***,
(90) v = a2 + b3(1 + ala2).
The only new syzygies needed for this reduction are
LG -Q2L2 + L6 =W, FLK = AW + AQ1 + (J + 1)K2,
(91) (F2 + L4 + Q2)K = (A + 1)L3,
(A + 1)(FG + KL4 + KQ2) + JKQ2 = ALQ1 + oL3,
in which w is an invariant not computed. Proof need not be
given of these facts since we presuppose below merely the existence of relation (89) which may be verified independently.
Of course, the fact that V is the only new covariant of rank 2
was a guide in the later investigation.
COVARIANTS OF EVEN RANK m = 2j > 0, ~~ 27-29
27. First, let n be odd. In the covariant (76) replace x3 by
x3 + xi. In view of (82), we get
' = f2,2m + f33m + fi'(2 + 32)r + (X1X2X3 + x12X2)4'.
Using the notation (84) for f2, we have S1' = S1 + S in f2.
Thus, as in ~ 17, S is a linear combination of the functions (74).
Now Q11Ln and its products by A and A + A are covariants (76)
with S given by (63). Using also KmL, in which S = a2(b3+1),
and its product by A, we may set
S = kl(b3 + a2) + k2b3ala2 + k3Aa2 + k4(b3 + a2)J.
In x1x234, let g be the coefficient of
XlX2X3 ' 222 —I134/z n —2 = (x22X34) 1x3n-1- 1