58 KANSAS UNIVERSITY QUARTERLY. 



in Salmon's Higher Plane Curves. But the method here sketched out 

 is very different in its point of view and much wider in its application, 

 yielding a multitude of new theorems not suggested by Cayley's meth- 

 od. 



TRINODAL QUARTICS. 



The quartic with three double points is a curve of the sixth class 

 having four double tangents and six cusps (Salmon's H. P. C. Art. 

 243). Hence its reciprocal is of the sixth degree with four double 

 points, six cusps, three double tangents, and no points of inflection. 



The locus of intersection of tangents to a conic at right angles to 

 one another is a circle. Inverting: — the locus of intersection of circles 

 through the node and tangent a nodal, bicircular quartic and at right 

 angles to one another is a circle. Projecting: — through the three nodes 

 of a quartic draw two conies, each touching the quartic and intersect- 

 ing so that the two tangents to the conies at their point of intersection, 

 together with the lines from it to two of the nodes, form a harmonic 

 pencil; the locus of all such intersctions is a conic through these two 

 nodes. Whenever the two tangents to the quartic from the third node, 

 together with the lines from it to the other two nodes, form a harmon- 

 ic pencil, this last conic breaks up into two right lines. 



Any chord of a conic through O is cut harmonically by the conic 

 and the polar of O. Inverting from O and projecting: — from one of the 

 nodes of a trinodal quartic draw the two tangents to the quartic (not 

 tangents at the node); draw the conic through these two points of con- 

 tact and the three nodes; any line through the first mentioned node is 

 cut harmonically by this conic, the quartic and the line joining the 

 other two nodes. 



If a triangle circumscribe a conic, the three lines from the angular 

 points of the triangle to the points of contact of the opposite sides in- 

 tersect in a point. Inverting and projecting: — through the three nodes 

 of a quartic draw three conies touching the quartic; through the point 

 of intersection of two of these conies, the point of contact of the 

 third, and the three nodes draw a conic; three such conies can be 

 drawn and they pass through a fixed point. 



The eight points of contact of two conies with their four common 

 tangents lie on a conic, which is the locus of a point, the pairs of tan- 

 gents from which to the two given conies form a harmonic pencil. In- 

 verting and projecting: — two connodal trinodal quartics have four com- 

 mon tangent conies through the three nodes; their eight points of con- 

 tact lie on another connodal trinodal quartic; if from any point on the 

 last quartic four conies be drawn through the nodes and tangent in 

 pairs to the first quartics, any line through a node is cut harmonically 

 by these four conies. 



