62 KANSAS UNIVERSITY QUARTERLY. 



of the same kind. These three quartics all have the same node, 

 cusps, and base conic. 



Every focal chord of a conic is cut harmonically by the curve, 

 the focus, and directrix. Inverting: — every nodal chord of a limacon 

 is bisected by the base circle. Projecting: — every nodal chord of a 

 nodal bicuspidal quartic is cut harmonically by the quartic, the base 

 conic, and the line joining the two cusps. Reciprocating: — from any 

 point on the double tangent of a nodal bicuspidal quartic draw the 

 other two tangents to the quartic and a line to the intersection of the 

 inflectional tangents; the fourth harmonic to these lines envelopes a 

 conic. 



Since the limacon is symmetrical with respect to the axis, it follows 

 that the two points of inflection are situated symmetrically with 

 respect to the axis. Hence the line joining the two points of inflec- 

 tion is parallel to the double tangent. Therefore by projection we 

 infer the following general theorem for the nodal bicuspidal quartic: 

 the line joining the two cusps, the line joining the two points of inflec- 

 tion, and the double tangent meet in a point. Also the fourth har- 

 monic points on each of these lines lie on a line through the node. 

 Reciprocating: — the point of intersection of the cuspidal tangents, the 

 the point of intersection of inflectional tangents, and the node all lie 

 on a right line. From the node draw a fourth harmonic to this right 

 line and the tangents at the node; draw a fourth line harmonic to this 

 right line and the inflectional tangents; draw a fourth harmonic to the 

 cuspidal tangents and this right line; these three lines all meet in a 

 point on the double tangent. 



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 

 tr-icuspidal 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 con- 

 tact of the double tangent being the foci. I shall call such a 

 point on the double tangent the harmonic point of the cuspidal tan- 

 gent. Since any two inflectional tangents of a nodal cubic meet on 

 the harmonic polar of the third point of inflection, it follows that any 



