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

~ 175]

INFLEXION POINTS AND TRIANGLES

331

coincide (at P) if and only if e=0. In that case P is called a
point of inflexion of F1 =0 and xi =0 the infiexion tangent
to F =0 at P.
Thus P is a point of inflexion of F1 =0 if and only if it is on
H1=0. Hence, by ~ 174, each intersection of a cubic curve
f=0 without a singular point with its Hessian curve h=0 is a
point of inflexion off= 0, and conversely.
There is certainly at least one intersection. For, by eliminating X3 between f=0 and h=0, we get a homogeneous
equation in xi and X2, having therefore at least one set of solutions x l, x'2. Then, for xi =x, x2=x'2, the equations f=0,
h=0 have at least one common root x=x'3. Thus (x'l, x'2,
X'3) is an intersection and therefore a point of inflexion of f=0.
Taking this point as a vertex (0, 0, 1) of a triangle of reference
and proceeding as before, we get F of type F1 with e=0. If
the coefficient d of x23 in F is zero, F has the factor xl. But,
if F = xiQ, the derivatives
aF = aQ aF aQ aF aQ
-=Q+x-l =Xi - x=
axi axi ax2 aX2 Ex3 ax3
all vanish at a point of intersection of x1=0, Q=0, whereas
F=0 has no singular point. Hence d#O. Taking d3x2 as a
new X2, and then adding a suitable multiple of xl to X2 to delete
the term with x22X1, we get
F =X32x +C, C= X23 +3bx2X12+aX13,
H=2xl C11 C2 _4x32C22= 72xl bx2+ax1 bx - 24x32x2.
C21 C22 bxi X2
Eliminating x32 between F=0, H =0, we get
Xi C
-X2 3x1(bx22 J+alxX2-b2Xl2)
= x24 +6bx22X2 +4aX2X13 - 3b2X4 = 0.
If xi=0, then x2=O and the intersection is (0, 0, 1). For the
remaining intersections, we may set xl = 1; then each root of

(6)

r4+6br2+4ar- 3b2 = 0