Colloquium publications.

INVARIANTS AND NUMBER THEORY.

97

Thus
S = x(A + 1 + A +- alo2 + b6b3 + bia2) + k4a2blb3
(97)
-+ g(A + 1 + A + alac2)b3 + Ik5Ab(b3 + 1).
First, let n = 4v + 2 and write 2g + 1 for m. Then
GQiLQ2v, K2GQil-1Q2v
have d = f3(f1i + 1) and a2d as the coefficients of 2mX3f. As
in ~ 25, the coefficients of ax, k4, g, k5 in (97) equal respectively
d + a2(A + b3 + 1), a2d + a2b, Ad + a2d + a2J, Ad.
The terms not containing d are combinations of the above a211
and a2(b3 + 1) of ~ 23. Any covariant with m = 2 - + 1 > 1,
n = 4 + 2, differs from one of rank > m by a linear function of
iKmLn, I1Km-1MQ2v, IGQlAQ2v, IK2GQl-1-Q2
(i= 1, A; = 1, A, J;I= 1, A, A).
Next, let n = 4v > 0. In the last two covariants of the
theorem below, the coefficients of t22'1+lx34 are a2b3(bl + 1) and
6 = b3j3(1 + 1). We had reached covariants in which the
corresponding coefficients are a2I and a2(b3 + 1)I. Thus we
obtain the coefficient of k4 in (97) and 8 + Aa2b3 + a2blb3, which
equals the coefficient of g. We may therefore set k4 = g = 0.
Subtracting covariants of the fourth and fifth types in the
theorem, we may take as S1 the function in ~ 24, without disturbing S. Applying (a1a2)(blb2) to S and S1, we get B, and
Bn-1. If I is the coefficient of Xlx2m+lX32'n+-2 in xlx2x34, its
coefficient in R' of ~ 30 is 1' = 1 + Bn + B,-1. Thus Bn + B,1
is of the form (57). By the coefficient of a3bl, t4=0. Since the
coefficient of a3 is zero for b3 = a2, we get x = k = t3 = 0.
Thus S= 0. Any covariant with m = 2M + I > 1, n = 4 > 0,
differs from one of rank > m by a linear function of
KImLn, AKmLn, IKmQ2V, iLn-3Q@+l, iLn-3K2+2, G2K@Q-IQQ-1,
FGQ2V-1Q1 (i = 1, A, A; I = 1, A, J).
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