Theory and applications of finite groups, by G.A. Miller, H. F. Blichfeldt [and] L. E. Dickson.

338 INFLEXION POINTS OF A CUBIC CURVE [Cu. XVIII

Consider the special cubic form given by F in ~ 175 with
a=l, b=-1. Then D=-48, so that P and each y, is irrational. Equation (6) is now
(6') r4- 6r2+4r - 3 = 0
and is irreducible in the domain R(1) of rational numbers.
For, no root is ~1, ~3, so that no root is rational. Further,
a y{ occurs in the coefficients of any quadratic factor, as shown
by Ferrari's method of solving quartic equations. Hence
the group of (6') for the domain R(1) is the symmetric group
(Ex. 5, ~ 153).
Let f be a cubic form whose ten coefficients are independent
variables. Let R be the domain of the rational functions with
rational coefficients of these ten variables. Then (Ex. 2,
~ 153), the group of the quartic equation (6) for R is the symmetric
group.
178. Group G of Equation X is the Linear Group L. After
the adjunction of a root r of the resolvent quartic (6), the
product of the equations of the three sides of an inflexion triangle has its coefficients in the domain (R, r), and the group G
reduces to the subgroup which permutes the triples of abscissas
of the points of inflexion on the sides of that triangle. First,
let the triangle be that one whose sides contain the points of
the rows in (10); these triples are merely permuted when the
sign of either index is changed and also by the transformation
='- +c, '-A ~+v+C
and hence by the group of the 4.33 transformations (12) with
b 0, whose index under L is 4. By the interchange of the two
indices, these triples are replaced by those in the columns of
(10), so that the latter are merely permuted by the group
of the transformations (12) with A —. When b=A-0, we
have
'=-a~+c, 7=l-Bi+C, aB;O (mod 3).
Unless a=B, the triples in the positive terms of the determinant
(10) are replaced by those in the negative terms, since this is
true for i'=, i' - -, and since each transformation
(15) 4'- +~+c, a'- -i+C