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.