54 KANSAS UNIVERSITY QUARTERLY. 



ent, which cuts every cubic of the system in three coincident points, 

 and hence is a common inflectional tangent. 



Since the polar conic of a point of inflection on a cubic consists of 

 the inflectional tangent and the harmonic polar of the point, and since 

 the polar conies of a fixed point with respect to a system of cubics 

 pass through four fixed points, it follows that in a system of cubics 

 having a common point of inflection and a common inflectional tangent 

 the harmonic polars of the common point of inflection meet in a point. 



Since the common inflectional tangent is the common polar line of 

 the common point of inflection, it follows that such a point is a critic 

 centre of the system of cubics. One cubic of the system then has a 

 node at the common point of inflection of the system, and forms an 

 exception. The line which is the common inflectional tangent to the 

 other cubics of the system cuts this also in three points, but is one of 

 the tangents at the double point; the other tangent at the double point 

 goes through the vertex of the pencil of harmonic polars. 



It is evident that the nine basal points of a system of conies may 

 unite into three groups of three each. The cubics will then all have 

 three common points of inflection, and at these points three common 

 inflectional tangents. These three points all lie on a line. 



When four basal points of the system of cubics coincide, such a 

 point is a double point on every cubic of the system. This is easily 

 shown as follows, using the method of inversion. Let a system of 

 conies Through four points be inverted from one of the four points. 

 The system of conies inverts into a system of cubics, having a common 

 node and passing through three other finite fixed points and the two 

 circular points at infinity. Since the common node counts as foiir 

 points of intersection, it follows that any two cubics of the system, and 

 hence all of them, intersect in nine points. This system can be pro- 

 jected into a system having a common double point and passing 

 through any five other fixed points. 



A number of theorems concerning the system of cubics can easily 

 be inferred from known theorems concerning the system of conies. 

 Since two conies of the system are parabolas, it follows that two cubics 

 of the system are cuspidal. Since three conies of the system break up 

 into pairs of right lines, it follows that three cubics of the system 

 break up into a right line and a conic. Each right line and its corres- 

 ponding conic intersect in the common double point. The line at 

 infinity cuts the system of conies in pairs of points in involution, the 

 points of contact of the two parabolas of the system being the foci; it 

 follows on inversion that the pairs of tangents to the cubics at their 



