Colloquium publications.

96

THE MADISON COLLOQUIUM.

from one of rank > m by a linear combination of
iMK-?Q2v, jKmLn, GKm-lQ2V, IFnl2Q,, jLnQl,, EQ1@Q2v-1
(i = 1, A, J; j = 1, A; I = 1, A, A, J).
COVARIANTS OF ODD RANK m = 2/A + 1 > 1, ~~ 30-31
30. Replacing x3 by x3 + x1 in the covariant (76), we get
R' = f2'2 + f323m + fi'(i + 3)m + (xix2x3 + x12x2)'.
In xlx2x3s, let g be the coefficient of (XlX22) (X22X3)m-lX2n. The
coefficient of the corresponding term of R' is g' g + B, where
B is that of x2" in fi. Hence B is of the form (57).
First, let n be odd. Then S1' = S1 + S under (50), so that
S is a linear combination of functions (74) with a2(b3 + 1) and
its product by A deleted (~ 23). Thus S is the sum of the terms
(95) after the first. Applying (a1a2a3)(blb2b3) to B, of the form
(57), we see that S is of the form (96). By these two results,
S = t(b3 + a2 + 3 b3ala2 + bA + bA + a2A).
If I is the coefficient of (X2X32)mX3U-lXl in x1XZX3q, that in R' is
l' = 1 + nB,. Hence, for n odd, Bn is of the form (57). Interchanging the subscripts 1, 2 in Bn, we get S. Thus the coefficient
of aa in S vanishes for b3 = al, so that t = O. Any covariant with
m and n odd differs from one of rank > m by a linear function of
KmLn and AKmL.
31. Finally, let m be odd and n even. According as n = 4v
or 4v + 2, KmQ2V or KIm-MQ2v is of the form (76) with a2 as
the coefficient of 62mX3n. Hence we may delete the terms a2I1
in (85) and hence the terms aljI in B of ~ 23. But (~ 30), B is
of the form (57). Now a3b1 occurs in J and AJ of I and in
b2J of b212, having in these linearly independent multipliers.
Hence
I = x(A+ )+ yA, I= e+fA+gA.
Since the coefficient of a3 in B shall vanish for b3 = a2, and B
itself if also al = 0, we get k1 = x = y = k8, k2 =f = = e.