68 KANSAS UNIVERSITY QUARTERLY. 



u, + (A+kA,)x2y+(B+kB0xy2 + (C+kC0y3 + (D+kD0xy+ 

 (E+kEj)y2+(F+kFjy=o 



There is one value of k for which the last term vanishes, and hence 

 the origin is a critic centre. The polar cubics of the point of undula- 

 tion break up into a system of conic& through four points and the 

 common tangent at the common point of undulation. For the equa- 

 of the polar cubics is 



y^ (A+kA0x2 + (B+kK0xy+(C+kC,)y^ + 2(D+kD,)x+ 

 2(E+kEJy+(F-fkF0 ^=o 



The system of polar conies of the origin consequently breaks up into 

 the line y=o and a pencil meeting in a point. The common tangent 

 at the common point of undulation is also the common polar line of 

 the point of undulation. 



When the four coincident basal points form a common double 

 point on the quartic, it is not difficult to show that two of the quartics 

 are cuspidal at this point. The polar cubics of the common double 

 point form a system having the same point for common double point. 

 The tangents to the quartics at the common node constitute the sys- 

 tem of polar conies and form a pencil in involution. Twelve of the 

 sixteen basal points may unite in three groups of four each and the 

 system of quartics is then trinodal and passes through four other fixed 

 points. This is the system obtained by inverting a system of conies 

 through four points and then projecting. 



A few special cases should be noticed here. If the four fixed points 

 and two of the nodes lie on a conic, this conic together with the two 

 lines from the third node to the first two constitute a quartic of the 

 system. If the four fixed points lie on a line, the quartic then consists 

 of this line and the sides of the triangle formed by the nodes. If the 

 three nodes and three of the fixed points lie on a conic, the system of 

 quartics then consists of this conic and a system of conies through the 

 three nodes and the fourth fixed point. A special case of a system of 

 quartics with three nodes is a system of cubics having a common node 

 and passing through five other fixed points together with a line through 

 two of them. 



If a fifth basal point be moved up to join the four at the common 

 node, the quartics have one tangent at the common node common to 

 all. If six basal points coincide they have both tangents at the node 

 common to all. In this case one of the quartics has a triple point at 

 the common node of the others. If seven basal points coincide, one of 

 these tangents is an inflectional tangent as well. If eight points coincide, 

 both are inflectional tangents. 



