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.