The quartic with three double points is a curve of the sixth class
having four double tangents and six cusps (Salmon’s H. P. C. Art. 243).
Hence its reciprocal is of the sixth degree with four double points,
six cusps, three double tangents, and no points of inflection.
The locus of intersection of tangents to a conic at right angles to
one another is a circle. Inverting:—the locus of intersection of
circles through the node and tangent a nodal, bicircular quartic and
at right angles to one another is a circle. Projecting:—through the
three nodes of a quartic draw two conics, each touching the quartic
and intersecting so that the two tangents to the conics at their point
of intersection, together with the lines from it to two of the nodes,
form a harmonic pencil; the locus of all such intersections is a conic
through these two nodes. Whenever the two tangents to the quartic from
the third node, together with the lines from it to the other two nodes,
form a harmonic pencil, this last conic breaks up into two right lines.
Any chord of a conic through O is cut harmonically by the conic and the
polar of O. Inverting from O and projecting:—from one of the nodes of
a trinodal quartic draw the two tangents to the quartic (not tangents
at the node); draw the conic through these two points of contact and
the three nodes; any line through the first mentioned node is cut
harmonically by this conic, the quartic and the line joining the other
two nodes.
If a triangle circumscribe a conic, the three lines from the angular
points of the triangle to the points of contact of the opposite sides
intersect in a point. Inverting and projecting:—through the three nodes
of a quartic draw three conics touching the quartic; through the point
of intersection of two of these conics, the point of contact of the
third, and the three nodes draw a conic; three such conics can be drawn
and they pass through a fixed point.
The eight points of contact of two conics with their four common
tangents lie on a conic, which is the locus of a point, the pairs of
tangents from which to the two given conics form a harmonic pencil.
Inverting and projecting:—two connodal trinodal quartics have four
common tangent conics through the three nodes; their eight points
of contact lie on another connodal trinodal quartic; if from any
point on the last quartic four conics be drawn through the nodes and
tangent in pairs to the first quartics, any line through a node is cut
harmonically by these four conics.
The eight common tangents to two conics at their common points all
touch a conic. Inverting and projecting:—two connodal trinodal quartics
intersect in four other points; eight conics can be drawn through the
three nodes tangent to the quartics at these points of intersection;
these eight conics all touch another connodal trinodal quartic.
A series of conics through four fixed points is cut by any transversal
in a range of points in involution. Inverting and projecting:—a series
of connodal trinodal quartics can be passed through four other fixed
points; any conic through the three nodes cuts the series of quartics
in pairs of points which determine at a node a pencil in involution.
The conic touches two of the quartics and the lines to the points of
contact are the foci of the pencil.
If the sides of two triangles touch a given conic, their six angular
points will lie on another conic. Inverting and projecting:—if two
groups of three conics each be passed through three nodes and tangent
to the quartic, their six points of intersection (three of each group)
lie on another connodal trinodal quartic.
If the two triangles are inscribed in a conic, their six sides touch
another conic. Inverting and projecting:—if two groups of three conics
each be passed through the three nodes of a quartic so that the three
points of intersection of each group lie on the quartic, these six
conics all touch another connodal trinodal quartic.
A triangle is circumscribed about one conic, and two of its angular
points are on a second conic; the locus of its third angular point is a
conic.—Inverting and projecting:—if three conics be drawn through the
three nodes of two connodal trinodal quartics so that they all touch
one of the quartics and two of their points of intersection are on the
other quartic, the locus of their third point of intersection is a
connodal trinodal quartic.
A triangle is inscribed in one conic and two of its sides touch a
second conic; the envelope of its third side is a conic. Inverting and
projecting:—if three conics be drawn through the three nodes of two
connodal trinodal quartics so that their three points of intersection
lie on one of the quartics and two of them touch the other quartic, the
envelope of the third conic is another connodal trinodal quartic.
The theorems of this section are stated in the most general terms and
are still true when one or more of the nodes are changed into cusps. It
is therefore not necessary to give separate theorems for the case of
one cusp and two nodes.