Colloquium publications.

42

THE MADISON COLLOQUIUM.

where a, b, c are arbitrary variables. Under the transformation
(21) x = x' y', y= y',
f becomes f', in which the coefficients are
(22) a'-a, b' -b, c' a+b +c (mod 2).
By ~ 3, the only invariants under d' - d, c' = c + d, modulo 2,
are the polynomials in d and c(c + d). Take d = a + b.
Hence the only seminvariants of f are the polynomials in a, b and
(23) s = c(c + a+ b).
Such a polynomial is an invariant of f if and only if it is
unaltered by the substitution (ac) induced by (xy). Thus
(24) b, k = as, q= b(a + c) + a2 + ac + c2 = s + ab + a2
are invariants of f. Introducing q in place of s, we see that any
seminvariant is a polynomial in a, b, q. Consider an invariant
of this type. Since its terms free of a are invariants, the sum
of its terms involving a is an invariant with the factor a and
hence also the factors c and a + b + c, the last by (22). Hence
this sum has the factor k, and its quotient by k is an invariant.
By induction we have the theorem:
Any rational integral formal invariant of f equals a rational
integral function* of b, q, k.
7. Formal Seminvariants of a Binary Quadratic Form for p > 2.
Write
(25) f = ax2 + 2bxy + cy2,
where a, b, c are arbitrary variables. Under the transformation
(21), f becomes f', whose coefficients are
(26) a' = a, b' = a + b, c' = a + 2b + c.
* Replace xl, X2, X3, of ~ 4 by a, b, c; then
L3 = bk(k + bq), Q32 = b4 + bk + q2, Q31 = b2q2 bqk + b3k + k2.