58 



^ 17. Ill order (o gel the eoinplex cone of an arbitrarj point l\ 

 we consider a plane :7 llirougli Ml*\ let () lie the intersection of' 

 MP, /) liie intersection of rr vvilli F^. If J\ and I\ correspond to 

 /}, PO, Pl\ and PP^ are the lines of intersection of the complex 

 cone of P with jt. If jt revolves round PO it appears that: 



the complex cone of a point P passes through the siniight lines 

 PM and PPij and touches the planes MPDi along the lines PDi. 



At the same time it appears again that if /* moves along a sti-aight 

 line through M, the curve at infinity of the complex cone of P 

 remains unaltered (cf. \ 8). 



§ 18. Out of a point 0, 4 real tangents ODi may be drawn to 

 the corresponding curve k^, hence the curves k'^ m\(\ £(\^o the com plex 

 cones consist of two paints. 



The caracteristic of a curve k^ is defined l\y the 4 straight lines 

 ODi. Through Di there pass 3 conies through the points of which 

 there pass 4 harmonical rays through Z)/, hence: 



tlie locus of the points loith harmonical complex cones consists of 

 3 quadratic cones the vertices of lohich lie in M and which pass 

 through the straight lines mi, and also : 



the complex cones of the points hjing on a quadratic cone through 

 the 4 straight lines nii, have the same characteristic. 



^19. The curve of Jacobi of the net of the curves k^ consists of 

 the six sides of the complete quadrangle of the points />)i. No rational 

 curves k^ belong to the net, only curves degenerated in a side of 

 the cpiadrangle and a conic through the 4 points Z),- and Hj that 

 do not lie on this side, accordingly : 



tliere are no points with rational complex cones; for any point of 

 a perpendicular bi.fector plane the complex cone degenerates into a 

 plane pencil and a quadratic cone; for a point V^ the complex cone 

 consists of a plane pencil in V^, to he counted double, and a single 

 plane pencil. 



§ 20. As each complex curve has a double tangent, we might 

 call those planes where the double tangent is an inflexional tangent, 

 singular planes. In this case the two points P con-esponding to the 

 straight line p in the null system [2,1], must coincide. This hap|)ens 

 only when a plane a passes through one of the points Di, but 

 then the system of complex rays in tt splits up into a plane pencil 

 and the tangents of a parabola; consequently non-degenerate complex 

 curves lüith an inflexional tangent do not occur. 



