On the Collinearity of Three Points on a Non-Singular Cubic

Every non-singular cubic can be reduced to the canonical form

x³ + y³ + z³ + 6mxyz = 0;

and if three points (xi, yi, zi) (i = 1,2,3) on this cubic be collinear, we have the well-known condition due to Cayley, viz.

x₁x₂x₃ + y₁y₂y₃ + z₁z₂z₃ = 0.

We have shown here that this condition is necessary but not sufficient.