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.
NODAL BICUSPIDAL QUARTICS.
A quartic with one node and two cusps is a curve of the fourth class, having one double tangent and two points of inflection (see Salmon). Hence its reciprocal is also a nodal bicuspidal quartic, a fact of which frequent note will be made in this section.