64 KANSAS UNIVERSITY QUARTERLY. 



the lines to the cusps is harmonic ; all such conies 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 inflec- 

 tional 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 conies is a 

 conic touching the three inflectional tangents. 



The directrix of a parabola is the locus of the intersection of tan- 

 gents 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 conies 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 inflec- 

 tional tangents to a nodal cubic draw the two tangents P and Q ; draw 

 two conies 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 c )mmon 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 para- 

 bola. 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 conies through the other two cusps and the extremities of each 

 of these chords so that the pole of the line joining the other two 



