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

3(a + ka₁) + (b + kb₁) x + (c + kc₁)y = 0 ,

which represents a pencil of lines through a point.

Suppose now the origin to be at one of the critic centres; then for a particular value, k₁, all terms lower than the second degree must vanish, so that

║abc║
║ ║ = 0.
║a₁  b₁  c₁║

The factors of the terms of u₂, which involves k₁, represent the tangents at the double point to the nodal cubic, and also the polar conic of the origin with respect to this nodal cubic. Hence a critic centre is at one of the vertices of the self-polar triangle of its system of polar conics. The opposite side of this triangle is the common polar line of the critic centre with respect to its system of polar conics, and hence it is also the common polar line of the critic centre with respect to the system of cubics. The four basal points of the system of polar conics lie two and two upon the tangents at the double point of the nodal cubic.

When the origin is taken at one of the nine basal points of the system of cubics, a and a₁ both vanish. Hence it is readily seen that a basal point of a system of cubics is also a basal point of its system of polar conics and the vertex of its pencil of polar lines.

Suppose two of the basal points of the system of cubics to coincide, then every cubic of the system, in order to pass through two coincident points, must touch a common tangent at a fixed point. The common tangent is the common polar of its point of contact, both with respect to the system of cubics and to its system of polar conics. Hence the union of two basal points gives rise to a critic centre. The self-polar triangle of its system of polar conics here reduces to a limited portion of the common tangent. This line is not a tangent to the nodal cubic, but only passes through its double point.

Suppose three of the basal points of a system of cubics to coincide, such a point will then be a point of inflection on each cubic of the system. For, in the last case, if a line be drawn from the point of contact of the common tangent to a third basal point of the system, such a line will be a common chord of the system of cubics. Suppose, now, this third basal point be moved along the curves until it coincides with the other two; then the common chord becomes a common tangent, which cuts every cubic of the system in three coincident points, and hence is a common inflectional tangent.

Since the polar conic of a point of inflection on a cubic consists of the inflectional tangent and the harmonic polar of the point, and since the polar conics of a fixed point with respect to a system of cubics pass through four fixed points, it follows that in a system of cubics having a common point of inflection and a common inflectional tangent the harmonic polars of the common point of inflection meet in a point.

Since the common inflectional tangent is the common polar line of the common point of inflection, it follows that such a point is a critic centre of the system of cubics. One cubic of the system then has a node at the common point of inflection of the system, and forms an exception. The line which is the common inflectional tangent to the other cubics of the system cuts this also in three points, but is one of the tangents at the double point; the other tangent at the double point goes through the vertex of the pencil of harmonic polars.

It is evident that the nine basal points of a system of conics may unite into three groups of three each. The cubics will then all have three common points of inflection, and at these points three common inflectional tangents. These three points all lie on a line.

When four basal points of the system of cubics coincide, such a point is a double point on every cubic of the system. This is easily shown as follows, using the method of inversion. Let a system of conics through four points be inverted from one of the four points. The system of conics inverts into a system of cubics, having a common node and passing through three other finite fixed points and the two circular points at infinity. Since the common node counts as four points of intersection, it follows that any two cubics of the system, and hence all of them, intersect in nine points. This system can be projected into a system having a common double point and passing through any five other fixed points.

A number of theorems concerning the system of cubics can easily be inferred from known theorems concerning the system of conics. Since two conics of the system are parabolas, it follows that two cubics of the system are cuspidal. Since three conics of the system break up into pairs of right lines, it follows that three cubics of the system break up into a right line and a conic. Each right line and its corresponding conic intersect in the common double point. The line at infinity cuts the system of conics in pairs of points in involution, the points of contact of the two parabolas of the system being the foci; it follows on inversion that the pairs of tangents to the cubics at their common node form a pencil in involution, the two cuspidal tangents being the foci.

If the four basal points of the system of conics lie on a circle, this circle inverts into a right line, and one cubic then consists of this right line and the lines joining the centre of inversion to the circular points at infinity. This theorem may be stated for the system of cubics as follows: if the conic determined by the five basal points of the system of cubics (not counting the common double points), break up into right lines, the line passing through three of the five points, together with the lines joining the other two points to the common node, constitute a cubic of the system.

If three of the four basal points of the system of conics lie on a line, the conics consist of this line and a pencil of lines through the fourth basal point. Inverting from this fourth point and then projecting, we have a system of cubics consisting of a pencil of lines and a conic through the vertex and the four other fixed points. Hence, when the five fixed points of such a system of cubics lie on a conic through the common node, this conic is a part of every cubic of the system. If we invert the above system of conics from one of the three points on the right line, and then project, we obtain a system of cubics which consists of a system of conics through four fixed points, and a fixed right line through one of these four points. Hence, if two of the five basal points of such a system of cubics be on a line through the common node, this line is a part of every cubic of the system.

If a system of conics having one basal point at infinity be inverted from one of the remaining basal points, this point at infinity inverts to the center of inversion, and we obtain a system of cubics having five coincident basal points and hence passing through only four others. The system of cubics is now so arranged that one tangent at their common double point is common to all. Only one cubic of the system is cuspidal. As before three cubics break up into a right line and conic.

If two of three basal points of the system of conics be at infinity, the system of cubics obtained by projection and inversion has six coincident basal points and hence only three others. This system has both tangents at the common node common to all cubics of the system. If the two basal points at infinity in the system of conics be coincident, all the conics are parabolas, and hence all the cubics of the system are cuspidal and have a common cuspidal tangent.

If three of the basal points of the system of conics be at infinity, the conics consist of the line at infinity and a pencil of lines through the finite basal point. Inverting from the latter, we obtain a system of cubics with seven coincident basal points. This system is made up of a pencil of lines meeting in the seven coincident basal points together with the two lines joining this to the other two basal points of the system. These two lines are part of every cubic of the system.

If one of the remaining basal points be moved up to join the seven coincident ones, one of these fixed lines becomes indeterminate, and the system of cubics through eight coincident points consists of a fixed line through the eight coincident points and the ninth fixed point together with any two lines of the pencil through the eight points. If the nine basal points coincide, any three lines through it form a cubic of the system.