newson: unicursal curves by method of inversion. 63 



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 conies through the three cusps of a tricus- 

 pidal quartic and touching the quartic intersect in three collinear 

 points. Reciprocating: — if three conies touch the three inflectional 

 tangents of a nodal cubic and the cubic itself, their three other com- 

 mon 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 conies, 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 



