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

~ 1791

REAL POINTS OF INFLEXION

341

product P-h+24rf of the linear functions which vanish at
the sides of an inflexion triangle has as its coefficients quantities in the enlarged domain. The determination of the linear
factors requires the solution of a cubic equation. Consider the
inflexion triangle associated with the rows in (10); after the
adjunction of the roots of the corresponding cubic equation,
the group permutes the roots [}v] in the same row. The only
transformations (15) having this property are ' I, '_?+C,
which form a group C3. The group of the resolvent cubic is
therefore of order. 18=6. In the new domain, the group of
the corresponding resolvent cubic for another inflexion triangle
is C3. After the adjunction of one and hence all of its roots,
we have the sides of two inflexion triangles, and their intersections give the nine inflexion points.
Hence the determination of the infiexion points of an arbitrary
cubic curve requires the extraction of a cube root and three square
roots to solve the resolvent quartic equation, then the extraction
of a square root and two cube roots to solve the two cubic equations
which determine the sides of two inflexion triangles. No one of
these three cube roots and four square roots can be avoided or expressed rationally in terms of the others.
179. Real Points of Inflexion. Let the coefficients of
the equation of the cubic curve be real. After a suitable choice
of axes, the nine abscissas of the points of inflexion are the nine
distinct roots of an equation with real coefficients (~ 176).
Hence at least one point of inflexion is real. The reduction to
the form F in ~ 175 can therefore be effected by a real transformation. By ~ 177 the discriminant of the real quartic
equation (6) is -27D2 and hence is negative. Thus * there
are two distinct real and two imaginary roots. One of the real
roots is positive and the other is negative, as shown by the
values - oo, 0, +o of the variable r. By use of the same
values we see that the slope of the curve y=x4+6br2+...
corresponding to (6), is positive at the point whose abscissa
is the positive root and negative at that with the negative root.
At the points of inflexion (1, r, is) the slope is -4s2, by the
* Dickson's Elementary Theory of Equations, p. 45, or Ex. 5, p. 101.