newson: unicursal curves by method of inversion. 55 



common node form a pencil in involution, the two cuspidal tangents 

 being the foci. 



If the four basal points of the system of conies lie on a circle, this 

 circle inverts into a right line, and one cubic then consists of this right 

 line and the lines joining the centre of inversion to the circular points 

 at infinity. This theorem may be stated for the system of cubics as 

 follows: if the conic determined by the five basal points of the system 

 of cubics (not counting the common double points), break up into right 

 lines, the line passing through three of the five points, together with 

 the lines joining the other two points to the common node, constitute a 

 cubic of the system. 



If three of the four basal points of the system of conies lie on a line, 

 the conies consist of this line and a pencil of lines through the fourth 

 basal point. Inverting from this fourth point and then projecting, we 

 have a system of cubics consisting of a pencil of lines and a conic 

 through the vertex and the four other fixed points. Hence, when the 

 five fixed points of such a system of cubics lie on a conic through the 

 common node, this conic is a part of every cubic of the system. If 

 we invert the above system of conies from one of the three points on 

 the right line, and then project, we obtain a system of cubics which 

 consists of a system of conies through four fixed points, and a fixed 

 right line through one of these four points. Hence, if two of the five 

 basal points of such a system of cubics be on a line through the com- 

 mon node, this line is a part of every cubic of the system. 



If a system of conies having one basal point at infinity be inverted 

 from one of the remaining basal points, this point at infinity inverts 

 to the center of inversion, and we obtain a system of cubics having 

 five coincident basal points and hence passing through only four others. 

 The system of cubics is now so arranged that one tangent at their 

 common double point is common to all. Only one cubic of the system 

 is cuspidal. As before three cubics break up into a right line and 

 conic. 



If two of three basal points of the system of conies be at infinity, the 

 system of cubics obtained by projection and inversion has six coinci- 

 dent basal points and hence only three others. This system has both tan- 

 gents at the common node common to all cubics of the system. If the 

 two basal points at infinity in the system of conies be coincident, all the 

 conies are parabolas, and hence all the cubics of the system are cus- 

 pidal and have a common cuspidal tangent. 



If three of the basal points of the system of conies be at infinity, the 

 conies consist of the line at infinity and a pencil of lines through the 

 finite basal point. Inverting from the latter, we obtain a system of cu- 



