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

~ 176]1

GROUP FOR THE INFLEXION POINTS

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r- p, we see by subtraction that F has the factors I and Q-ll12
and hence has a singular point, contrary to assumption.
Corresponding results follow at once for the general cubic
curve f=0. We saw that f can be reduced to F by a linear
transformation of a certain determinant 8. But F is replaced
by a like form by the transformation which multiplies xi,
X2, X3 by 5-2, 1,. 6, respectively, and thus has the determinant
8-1. The product of the two transformations is of determinant unity and replacesf by a form F. Hence (~ 174), it replaces
the Hessian h of f by the Hessian H of F. Thus for each root
r of (6), in which a and b are certain functions of the coefficients
of f, h+24rf=0 represents an inflexion triangle of f.
Furthermore, a and b are rational functions of the coefficients
off. For, there are exactly four values of r for which _ h + 24rf
has a linear factor xl-mx2-nX3. Replacing xi by mx2+fnx3
in 0, we obtain a cubic function of X2 and X3 whose coefficients
must vanish. Eliminating m and n, we obtain two equations
in which r and the coefficients of f enter rationally and integrally. The greatest common divisor of their left members
must be a function of r whose coefficients are rational in those
off. The latter is therefore true (~ 145, first foot-note) of the
quartic equation * for r with no multiple root.
176. Group G of the Equation X for the Abscissas of the
Points of Inflexion. Let R be the domain defined by the coefficients of the equation f=0 of a cubic curve without singular
points. We employ a new triangle of reference whose side
xi=0 does not contain a point of inflexion. This can be accomplished by a linear transformation on xi, X2, X3 with coefficients in R. We pass to Cartesian coordinates by setting
X2/x==x, X3/xi = y. After applying a transformation with
coefficients in R, corresponding to a rotation of the axes, we may
assume that the y-axis is not parallel to any line joining two
inflexion points of f=0. Then the abscissas xi,..., X9 of
the points of inflexion are distinct. By eliminating y3 and y2
between the equations of the curve and its Hessian curve, we
* We do not employ the fact, which now follows readily, that the coefficients
of (6) are rational integral invariants of f.