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

332 INFLEXION POINTS OF A CUBIC CURVE [CH. XVIII

leads to the inflexion points (1, r, ts), where
-s2 =C=r3+3br+a.
If s=0, (6) would have a double root and a2+4b3=0. But
the partial derivatives of F all vanish at (1, x2, 0) if x22-+b =0,
2bx2+a=O, and hence if b=0, x2=0, or if b5O, x2=-a/(2b),
whereas F has no singular point. Hence
(7) a2+4b3~O
and (6) has four distinct roots r for each of which s 0. Thus
there are exactly nine distinct points of inflexion.
The two points (1, r, ~s) with a fixed r are collinear with
P= (0, 0, 1), being on x2=rxl. For the remaining roots p of
(6), we have
p3+rp2+(r2+6b)p+r3+6br+4a=0.
The product of this by r can be written in the form
r(p3 +3bp +a) + (rp+k)2 =, k-=(r2+3b).
Hence the quadratic factor in
(8) - 1H+rF(=(rXl-X2)2X32 -l(rx2+kxl)22
24 r
vanishes at (1, p, ~io), where
- 2= p3 +3bp+a.
Thus the nine points of inflexion lie by threes upon the three
straight lines given by (8), which are said to form an inflexion
triangle. There are four inflexion triangles, one for each root
r of (6).
The roots of (6) are the only values of r for which H+24rF
has a linear factor 1. In fact, I=0 meets F =0 in three points
on H =0 which are therefore points of inflexion. Thus I has two
companion linear functions such that 1112 =0 is one of the
four inflexion triangles. Hence 11112=H+24pF, where p is
one of the roots of (6). By hypothesis, IQ= H+24rF. If