NEWSON: UNICURSAL CURVES BY METHOD OF INVERSION. 59 



The eight common tangents to two conies at their common points 

 all touch a conic. Inverting and projecting: — two connodal trinodal 

 quartics intersect in four other points; eight conies can be drawn through 

 the three nodes tangent to the quartics at these points of intersection; 

 these eight conies all touch another connodal trinodal quartic. 



A series of conies through four fixed points is cut by any transver- 

 sal in a range of points in involution. Inverting and projecting: — a series 

 of connodal trinodal quartics can be passed through four other fixed 

 points; any conic through the three nodes cuts the series of quartics in 

 pairs of points which determine at a node a pencil in involution. The 

 conic touches two of the quartics and the lines to the points of contact 

 are the foci of the pencil. 



If the sides of two triangles touch a given conic, their six angular 

 points will lie on another conic. Inverting and projecting: — if two 

 groups of three conies each be passed through three nodes and tangent 

 to the quartic, their six points of intersection (three of each group) 

 lie on another connodal trinodal quartic. 



If the two triangles are inscribed in a conic, their six sides touch 

 another conic. Inverting and projecting: — if two groups of three con- 

 ies each be passed through the three nodes of a quartic so that the three 

 points of intersection of each group lie on the quartic, these six conies 

 all touch another connodal trinodal quartic. 



A triangle is circumscribed about one conic, and two of its angular 

 points are on a second conic; the locus of its third angular point is a 

 conic. — Inverting and projecting: — if three conies be drawn through 

 the three nodes of two connodal trinodal quartics so that they all touch 

 one of the quartics and two of their points of intersection are on the 

 other quartic, the locus of their third point of intersection is a conno- 

 dal trinodal quartic. 



A triangle is inscribed in one conic and two of its sides touch a sec- 

 ond conic; the envelope of its third side is a conic. Inverting and pro- 

 jecting: — if three conies be drawn through the three nodes of two con- 

 nodal trinodal quartics so that their three points of intersection lie on 

 one of the quartics and two of them touch the other quartic, the envel- 

 ope of the third conic is another connodal trinodal quartic. 



The theorems of this section are stated in the most general terms and 

 are still true when one or more of the nodes are changed into cusps. 

 It is therefore not necessary to give separate theorems for the case of 

 one cusp and two nodes. 



NODAL P.ICUSPIDAL QUARTICS. 



A quartic with one node and two cusps is a curve of the fourth 

 class, having one double tangent and two points of inflection (see 



