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
║ a b c ║
║ ║ = 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.