(7) Reciprocating (6):—from any point on the cuspidal tangent draw two other tangents, B and C, to the cubic; take any two other tangents, P and Q, such that the range cut from the inflectional tangent by B, C, P, Q, is harmonic; the conic touching B, C, P, Q, and the inflectional tangent will also touch the line joining the point of inflection and the cusp.
(8) Projecting (3):—through the point of inflection draw any line cutting the cubic in B and C; through the cusp O and the points B and C on the cubic and any other fixed point P, three, and only three, conics can be passed, such that the tangent to the conic and cubic at their remaining point of intersection, together with the lines from it to B and C, form a harmonic pencil; these three points of intersection are collinear.
SYSTEMS OF CUBICS THROUGH NINE POINTS.
Let U and V be the equations of two given cubics, then U + kV is the equation of a system of cubics through their nine points of intersection. Twelve cubics of this system are unicursal, and the twelve nodes are called the twelve critic centres of the system. (See Salmon’s H. P. C., Art. 190.)
Let the equation of the system be written briefly
a + ka₁ + (b + kb₁) x + (c + kc₁) y + u₂ + u₃ = 0 ;
one, and only one, value of k makes the absolute term vanish; hence one, and only one, curve of the system passes through the origin, which may be any point in the plane. Make the equation of the system homogeneous by means of z, and differentiate twice with respect to z; we obtain thus the equations of the polar conics and polar lines of the origin with respect to the system.
The polar conics of the origin are given by
3(a + ka₁) + 2{ (b + kb₁) x + (c + kc₁)y } + u₂ = 0 ;
thus showing that the polar conics of any point, with respect to the system of cubics, form a system through four points. The polar lines of the origin are given by