Colloquium publications.

INVARIANTS AND NUMBER THEORY.

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2. The Classes of Algebraic Quartic Forms.-Consider a quartic
form f in which ak is the first non-vanishing coefficient. Apply
transformation (1) with
(2) - ak+t
(k - +1)a'
We obtain a form having zero in place of the former ak+. Dropping the accents on x', y', we obtain, for k = 0, 1, 2, 3, the respective forms
(3) ao + 0: aox4 + 6ao-'S2xy2 + 4ao-2Sxy3 + ao3S4y4,
(4) ao = 0, al + 0: 4axzay + a-1513xy3 + a-2Sl4y4,
(5) ao = al = 0, a2 0: 6a2X2y2 + -a21S24y4,
(6) ao = a = a2= 0, as 0: 4a3xy3,
(7) ao = a= a2= a3 = 0: a4y4,
no transformation having been made in the last case. Here
(8) S2 = aoa2 - al2, S3 = ao2a3 - 3aoala2 + 2a13,
(9) S4 = ao3a4- 4ao2ala3 + 6aoal2a2 - 3a14,
S13 = 4ala3 - 3a22, S14 = a2a4 - 2aa2a3 + a23,
(10)
S24 = 3a2a4- 2a32.
If we apply to one of the forms (3)-(6) a transformation (1)
with t = 0, we obtain a form having an additional (second)
term. Hence no two of the forms (3)-(7) can be transformed
into each other by a transformation (1), so that each represents
a class of forms. For example, there is a class (5) for each set
of values of the parameters a2 and S24 (a2 + 0).
3. Rational Integral Seminvariants of an Algebraic Quartic.First, ao is a seminvariant since it has a definite value = 0 for
any form in any class (3) and the value zero for any form in
any class (4)-(7). Next, S2, S3, S4 are seminvariants, since
they have constant values
(11) S2= - al2, S3 = 2a13, S4 - 3a14 (if ao = 0)
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