The Kansas University Quarterly, Vol. I, No. 2, October 1892

by Various

TRICUSPIDAL QUARTICS.

A tricuspidal quartic is a curve of the third class with one double tangent and no inflection. Its reciprocal is therefore a nodal cubic.

We shall begin by reciprocating some of the simpler properties of nodal cubics. Since the three points of inflection of a nodal cubic lie on a right line, it follows that the three cuspidal tangents of a tricuspidal quartic meet in a point. The reciprocal of the harmonic polar of a point of inflection is a point on the double tangent, found by drawing through the point of intersection of the three cuspidal tangents a line forming with them a harmonic pencil. Three such lines can be drawn and it is not difficult to distinguish them. All six lines form a pencil in involution, the lines to the points of contact of the double tangent being the foci. I shall call such a point on the double tangent the harmonic point of the cuspidal tangent. Since any two inflectional tangents of a nodal cubic meet on the harmonic polar of the third point of inflection, it follows that any two cusps of a trinodal quartic and the harmonic point of the third cuspidal tangent lie on a right line. Since the point of contact of the tangents from a point of inflection of a nodal cubic is on the harmonic polar of the point, it follows that the tangent to the tricuspidal quartic at the point where it is cut by a cuspidal tangent passes through the harmonic point of that cuspidal tangent.

The inverse of the parabola from a focus is the cardioid; and the inverse of the corresponding directrix is the base circle of the cardioid. The cardioid projects into a tricuspidal quartic and its base circle projects into a conic through the three cusps which has the same general properties as the base conic of the nodal bicuspidal quartic.

The circle circumscribing the triangle formed by the three tangents to a parabola passes through the focus. Inverting:—three circles through the cusp, and tangent to a cardioid, intersect in three collinear points. Projecting:—three conics through the three cusps of a tricuspidal quartic and touching the quartic intersect in three collinear points. Reciprocating:—if three conics touch the three inflectional tangents of a nodal cubic and the cubic itself, their three other common tangents intersect in a point.

Circles described on the focal radii of a parabola as diameters touch the tangent through the vertex. Inverting and projecting:—from a point on a tricuspidal quartic lines are drawn to the three cusps and a fourth line forming a harmonic pencil; the envelope of this fourth line is a conic through the three cusps and touching the quartic at the point where the latter is cut by one of the cuspidal tangents. There are three such conics, one corresponding to each cusp. At any cusp the tangent to its corresponding base conic, the cuspidal tangent, and the lines to the other two cusps form a harmonic pencil. Reciprocating:—on any tangent to a nodal cubic take the three points of intersection with the inflectional tangents and a fourth point forming with these a harmonic range; the locus of this fourth point is a conic touching the three inflectional tangents and the cubic. The tangent to the cubic where it is touched by the conic goes through a point of inflection. On any inflectional tangent the point of contact of this conic, the point of inflection, and the points of intersection of the other two inflectional tangents form a harmonic range.

The circle described on any focal chord of a parabola as diameter will touch the directrix. Inverting:—the circle described on any cuspidal chord of a cardioid will touch the base circle. Projecting:—through a cusp C draw any chord of a tricuspidal quartic meeting the quartic in P and O; draw a conic through P, O, and the other two cusps so that the pencil at P formed by the tangent to the conic and the lines to the cusps is harmonic; all such conics will touch the base conic of the cusp C. Reciprocating:—from O, on any inflectional tangent of a nodal cubic, draw two tangents P and Q to the cubic; draw a conic touching the tangents P and Q and the other two inflectional tangents so that the range on one of these tangents formed by the point of contact of the conic and the intersection of the three inflectional tangent is harmonic; the envelope of all such conics is a conic touching the three inflectional tangents.

The directrix of a parabola is the locus of the intersection of tangents at right angles to one another. Inverting and projecting:—through any point P on the base conic of a cusp C of the tricuspidal quartic, two conics can be drawn through the three cusps and touching the quartic; their two tangents at P and the lines to the other two cusps form a harmonic pencil; their two points of contact lie on a line through C. Reciprocating:—from any point on one of the inflectional tangents to a nodal cubic draw the two tangents P and Q; draw two conics each touching the cubic and the three inflectional tangents, one touching P and the other Q; the envelope of their other common tangent is a conic touching the three inflectional tangents; the two points of contact of any one of these common tangents and the points where it cuts the other two inflectional tangents form a harmonic range.

Any two parabolas which have a common focus and their axes in opposite directions cut at right angles. Inverting:—any two cardioids having a common cusp and their axes in opposite directions cut at right angles. Projecting:—two tricuspidal quartics having common cusps and at one of the cusps the same cuspidal tangent, but the cusps pointed in opposite directions, cut at such an angle that the tangents at a point of intersection and the lines to the other two cusps form a harmonic pencil. Reciprocating:—two nodal cubics have common inflectional tangents and on one of them the points of inflection common, but the branches of the curve on opposite sides of the line; any common tangent to the two curves is cut harmonically by the points of contact and the other two inflectional tangents.

Circles are described on any two focal chords of a parabola as diameters; their common chord goes through the vertex of the parabola. Inverting:—circles are described on any two cuspidal chords of a cardioid; the circle through their points of intersection and the cusp goes also through the vertex of the cardioid. Projecting:—through one of the cusps of a tricuspidal quartic draw two chords; draw conics through the other two cusps and the extremities of each of these chords so that the pole of the line joining the other two cusps with respect to each of these conics is on the corresponding chord; the conic through the points of intersection of these two conics and the cusps passes also through the point where the cuspidal tangent of the first mentioned cusp cuts the quartic. Reciprocating:—on one of the inflectional tangents, of a nodal cubic take two points P and Q; draw a pair of tangents from each of these points to the cubic; draw two conics each touching a pair of these tangents and the other two inflectional tangents, so that the polars of the point of intersection of the other two inflectional tangents with respect to each of those conics pass respectively through P and Q; the conic touching the common tangents to these two conics and the three inflectional tangents touches also the tangent from the first mentioned point of inflection to the cubic.