Colloquium publications.

INVARIANTS AND NUMBER THEORY.

93

Such a term occurs in neither of the first two parts of K', since
they are functions of only two variables. To obtain such a
term from the third part of R', we must omit terms with the
factor $32 (and hence x12) and take (x2x32)2A in 12~, so as not to
make the degree in x2 too high. Hence if T be the coefficient of
x3n in fi, g' g + T. Now (ala2) (bb2) replaces S by T. The
resulting T must be of the form (57). By the coefficient of
a3bl, kc4 0; cf. (72). By the coefficient k3alb3 of a3, s 3- 0.
Since T = 0 for al = O, b3 = a2, we get k1 = k2. Hence S = kv,
where v is given by (90).
For n = 1, f2 = SX3 + S1X1. Thus Si = klv', where v' is
derived from v by interchanging the subscripts 1 and 3. Then
S1'=- S + S gives k1 0.
For n > 3, Qi-1L'-3V is of the form (76) with S = v, since
3v = 0.
Any covariant with n odd, m = 2g > 0, differs from one of rank
> m by a linear combination of IQiL (I = 1, A, A), KmL,
AKmL' and, if n > 1, Qjl-1LL-3V.
28. For m = 2i/ > 0, n = 4v > 0, the coefficients of P2mX3n in
) Q1Q2, KQ, Q1F2, Q1Ln, KImLn,
Q()-1 Q2QV-1G2, Km-S2Q2V-1G2
are respectively
1, a2, b3, f3 + 1, a2(b3 + 1), d= f3(/3 + 1), a2d.
These may be multiplied by any invariant. Now
t3 + 1 + a2 + b3 = a2,
A(f3 + 1) + (A + A + 1)b3 + b3a2 + A = b3la2,
d + A + a3 (+ b3) = bl(b3 + -2) -,
a2d + a2b3 = a2blb3 = a23, Ad = Abl(b3 + 1) = Aj.
Hence we have a covariant (76) in which the coefficient of x2mXan
is any linear combination of the functions (71). Hence the