u₂ + 3u₁ + 6u₀ = 0;
and the polar lines of the origin, by u₁ + 4u₀ = 0.
The origin may be any point in the plane and hence we conclude that
only one quartic of the system passes through a given point and that
the polar cubics of any point form a system through nine points. The
polar conics of any point form a system through four points and the
polar lines meet in a point.
If one of the critic centres be taken for origin, we can readily see
that such a point is also a critic centre on each of its systems of
polar curves. It is thus at a vertex of the self-polar triangle of
its system of polar conics and the opposite side of the triangle is
the common polar line of the critic centre with respect to each of
the systems of curves. The tangents at the node of the nodal quartic
coincide with those of its polar cubic and these we know coincide with
the lines which constitute its polar conic.
If two of the sixteen basal points coincide, such a point is a critic
centre. The argument is the same as for a system of cubics. We can also
see that two of the basal points of each of its systems of polar curves
coincide at the critic centre. The sixteen basal points of the system
of quartics may unite two and two so that it is possible to draw a
system of quartics touching eight given lines each at a fixed point.
If three of the basal points of our system of quartics coincide, all
the quartics have at such a point a common point of inflection and a
common inflectional tangent. The demonstration is the same as that
already given for cubics. The system of polar cubics of such a point
also have this point for a common point if inflection and the same
tangent for a common inflectional tangent. I prefer to show this
analytically for the sake of the method. The equation of the system of
quartics having the origin for a common point of inflection and the
axis of y for a common inflectional tangent may be written
u₄ + u₃ + {(B + kB₁)xy + (C + kC₁)y²} + (A + kA₁)y = 0.
The equation of the polar cubics of the origin is therefore,
u₃ + 2{(B + kB₁)xy + (C + kC₁)y²} + 3(A + kA₁)y = 0,
which proves the proposition. The properties of the system of polar
conics of such a point are therefore the same as those already proved
for cubics. One quartic of the system has a double point at the common
point of inflection of the others.
When four basal points coincide they give rise either to a common point
of undulation or a common double point on all the quartics of the
system. The equation of the system having a common point of undulation
may be written
u₄ + (A + kA₁)x²y + (B + kB₁)xy² + (C + kC₁)y³
+ (D + kD₁)xy + (E + kE₁)y² + (F + kF₁)y = 0.
There is one value of k for which the last term vanishes, and hence the
origin is a critic centre. The polar cubics of the point of undulation
break up into a system of conics through four points and the common
tangent at the common point of undulation. For the equation of the
polar cubics is
y{(A + kA₁)x² + (B + kK₁)xy + (C + kC₁)y²
+ 2(D + kD₁)x + 2(E + kE₁)y + (F + kF₁)} = 0.
The system of polar conics of the origin consequently breaks up into
the line y = 0 and a pencil meeting in a point. The common tangent at
the common point of undulation is also the common polar line of the
point of undulation.
When the four coincident basal points form a common double point on
the quartic, it is not difficult to show that two of the quartics are
cuspidal at this point. The polar cubics of the common double point
form a system having the same point for common double point. The
tangents to the quartics at the common node constitute the system of
polar conics and form a pencil in involution. Twelve of the sixteen
basal points may unite in three groups of four each and the system of
quartics is then trinodal and passes through four other fixed points.
This is the system obtained by inverting a system of conics through
four points and then projecting.
A few special cases should be noticed here. If the four fixed points
and two of the nodes lie on a conic, this conic together with the two
lines from the third node to the first two constitute a quartic of
the system. If the four fixed points lie on a line, the quartic then
consists of this line and the sides of the triangle formed by the
nodes. If the three nodes and three of the fixed points lie on a conic,
the system of quartics then consists of this conic and a system of
conics through the three nodes and the fourth fixed point. A special
case of a system of quartics with three nodes is a system of cubics
having a common node and passing through five other fixed points
together with a line through two of them.
If a fifth basal point be moved up to join the four at the common node,
the quartics have one tangent at the common node common to all. If
six basal points coincide they have both tangents at the node common
to all. In this case one of the quartics has a triple point at the
common node of the others. If seven basal points coincide, one of these
tangents is an inflectional tangent as well. If eight points coincide,
both are inflectional tangents.
When nine of the basal points of a system of quartics coincide, the
quartics have a common triple point. This is nicely shown by inverting
a system of nodal cubics from the common node. The inverse curves form
a system of quartics having a triple point and passing through seven
other fixed points. The common triple point on two quartics counts for
nine points of intersection and the seven others make the requisite
sixteen. From our knowledge of a system of cubics having a common node
it is readily inferred that three of the quartics must each break up
into a nodal cubic and a right line through the node. If the seven
fixed points of the system of quartics lie on a cubic having a node
at the common triple point, the system of quartics then consists of
this cubic and a pencil of lines through the node. If two of the seven
fixed points lie on a line through the common triple point, the system
of quartics then consists of this right line and a system of cubics
through the other five points and having a common node at the common
triple point.
The system of cubics having a common node may have one, two, or three
of the other basal points at infinity; and these may be all distinct
or two or three of them coincident. Whence we infer that if the system
of quartics have ten coincident basal points, one of the tangents at
the triple point is common to all the quartics of the system. If eleven
basal points coincide, two of the triple-point tangents are common to
all the quartics. If twelve coincide, all three triple-point tangents
are common. These triple-point tangents may be all distinct, two
coincident, or all three coincident.
If thirteen basal points coincide, the system of quartics then consists
of the three fixed lines joining the multiple point to the other three,
together with a pencil of lines through the multiple point. If fourteen
points coincide, two lines are fixed and these with any two lines of
the pencil form a quartic of the system. If fifteen points coincide,
only one line is fixed and each quartic consists of this line and any
other three of the pencil. When all sixteen points coincide, any four
lines through it form a quartic of the system.
In this paper cubic and quartic curves only are considered. I expect
in a future paper to extend the methods herein developed to curves of
still higher degrees. Many of the present results can be generalized
and stated for a unicursal curve of the nth degree. I have purposely
omitted all consideration of focal properties of these curves. There
are also many special forms of interest which do not properly belong to
a general treatment of the subject.