An elementary treatise on cubic and quartic curves, by A. B. Basset.

,POINTS OF INFLEXION.

61

The theorems of the last two articles show that the six
imaginary points of inflexion of an anautotomic cubic form three
conjugate pairs, and that a real straight line can be drawn
through any conjugate pair and one of the real points of inflexion.
It may be added that a pair of conjugate imaginary points are
such that the equations of the lines joining them to any vertex
(say A) of the triangle of reference are /3 + lky = 0, so that both
lines are included in the equation /32 + y27 = 0.
98. An acnodal cubic has three real points of inflexion, and a
crunodal cubic has one real and two inmaginary ones.
We have shown from Plicker's equations that a nodal cubic
cannot have more than three points of inflexion. Let A be the
node, C the real point of infiexion, BC the tangent at C. Then
the equation of the cubic is
/33 + (1,2 + 2m/,8 +~ ny/) a = 0............... (12).
Let B' be another point of inflexion, and let B'C' the tangent
at B' meet AC in C'. Then if / + kr = 0 and Xa +,// + vy = 0 be
the equations of AB' and B'C', the equation of the cubic must be
(/ + kcy) + (132 + 2r/fr7 + nry2) (Xa + LIM + rvy)= 0...(13).
In order that (12) and (13) should represent the same curve
we must have
k3 + 2n = 0,
3k2 + nq + 2mnv = 0,
3/c + Iv + 2mn O = 0.
Eliminating /u and v, we obtain
k {(4m2 - in) k2 - 6mnk -+- 3n2} = 0.
The solution k = 0 shows that C is a real point of inflexion,
whilst the quadratic factor gives the values of k for the lines
joining A to the other two points of inflexion. The condition
that these two lines should be real is that In > m2, and consequently the nodal tangents are imaginary or real according
as the other two points of inflexion are real or imaginary.
It frequently happens that when a cubic is drawn the number
of real points of inflexion is apparently defective. Whenever
this is the case, such singularities exist at infinity which can be
found by the methods of %~ 47 to 51.