If an ellipse be inverted from one of its vertices, the inverse
curve is symmetrical with respect to the axis; it has one point of
inflection at infinity and the asymptote is an inflectional tangent.
This asymptote is the inverse of the circle of curvature at the vertex.
The cubic has two other points of inflection situated symmetrically
with respect to the axis. Hence the three points of inflection lie on
a right line, a projective theorem which is consequently true of all
nodal cubics. The axis is evidently the harmonic polar of the point
of inflection at infinity. Since the axis bisects the angle between
the tangents at the node, it follows that the line joining a point of
inflection to the node, the two tangents at the node, and the harmonic
polar of the point of inflection, form a harmonic pencil. There are
three such lines, one to each node, and three harmonic polars; these
form a pencil in involution, the tangents at the node being the foci.
Since the asymptote is perpendicular to the axis, we have by projection
the following theorem:—through a point of inflection I, draw any line
cutting the cubic in B and C. Through P the point of intersection of
the harmonic polar and inflectional tangent of I, draw two lines to B
and C. The four lines meeting in P form a harmonic pencil. The point
of contact of the tangent from I to the cubic is on the harmonic polar
of I. Any two inflectional tangents meet on the harmonic polar of the
third point of inflection.
The locus of the foot of the perpendicular from the focus of a conic
on a tangent is the auxiliary circle. Inverting from the vertex, there
are two points, A and B, on the axis of the curve, such that if a
circle be drawn through one of them and the node, cutting at right
angles a tangent circle through the node, their point of intersection
will be on the tangent to the curve where it is cut by the axis.
Projecting:—through a point of inflection I of a nodal cubic draw a
line cutting the cubic in P and Q; there are two determinate points
on the harmonic polar of I, which have the following property:—draw a
conic through P, Q, and the node touching the cubic; draw another conic
through one of these points, P, Q, and the node cutting the former, so
that their tangents at their point of intersection, together with the
lines from it to P and Q form a harmonic pencil; the locus of such a
point of intersection is the tangent from I to the cubic.
If three conics circumscribe the same quadrilateral, the common tangent
to any two is cut harmonically by the third. Inverting from one of the
vertices of the quadrilateral: if three nodal, circular cubics have
a common double point and pass through three other fixed points, the
common tangent circle through the common node to any two of the cubics
is cut harmonically by the third; _i. e._, so that the pencil from the
node to the two points of intersection and the points of contact is
harmonic. Projecting this:—given three nodal cubics having a common
node and passing through five other fixed points; let a conic be passed
through the common node and two of the fixed points, touching two of
the cubics. The pencil from the common node to the points of contact
and the point where the conic cuts the third cubic is harmonic.
The following theorem may be proved in similar manner:—given a system
of cubics having a common node and passing through five other fixed
points; let a conic be drawn through the common node and two of the
fixed points; the lines drawn from the points where it cuts the cubics
to the common node form a pencil in involution.
A variable chord drawn through a fixed point P to a conic subtends a
pencil in involution at any point O on the conic. Inverting from O:—a
system of circles through the double point of a nodal circular cubic
and any other fixed point P, is cut by the cubic in pairs of points
which determine at the node a pencil in involution. Projecting:—a
system of conics through the node of a unicursal cubic, two fixed
points on the curve, and any fourth fixed point, is cut by the cubic in
pairs of points which determine at the node a pencil in involution.
We give another proof of the theorem that the three points of
inflection of a nodal cubic lie on a right line. This is easily shown
by inversion and is a beautiful example of the method.
There are three points on a conic whose osculating circles pass through
a given point on the conic; these three points lie on a circle passing
through the given point.[1] (Salmon’s Conics, Art. 244, Ex. 5.) By
inverting from the given point and then projecting, we readily see that
there are three points of inflection on a nodal cubic which lie on a
right line. If the above conic be an ellipse, the three osculating
circles are all real; but if it be a hyperbola, one only is real. Hence
an acnodal cubic has three real points of inflection, while a crunodal
one has one real and two imaginary.
The reciprocals of many of the theorems of this section are of interest
and will be given under Quartics.