56 



§ 10. The complex carüs of an aröitra>-(/ plane :x is ratioudl ; the 

 C ^\f ^^•"'' pl'^ne is its hi-tangent ; single tangents are: the lines of 

 intersection of Jt ?oith the jteypendicular bisector planes of T. 



Tliroiiji^li its l)i-tan^eiü, and tlie six single tangents tlie complex 

 curv^e of an arbitrary plane is defined; oilier tangeiris may be con- 

 structed with tlie ruler. 



§ \\. If the telrahedon T is cut by V^, we gel the well known 

 configuration of a complete quadrilateral. Polarisation of this figure 

 in the absolute polar field gives a complete quadrangle having D 

 as angular points; the straight lines at infinity of the perpendicular 

 bisectoi' planes are the sides and the points H^ are the diagonal 

 points of this quadi-angle. 



§ 12. In a plane n through one of the points Hj, hence parallel 

 to a normal to 2 subtending sides of 1\ the complex i-ays consist of 

 a plane pencil round Hj and the tangents of a parabola. If nr passes 

 at the same time through ]\1, it contains also a plane pencil round 

 M, hence also a third plane pencil; as the /^ of ti is a double ray 

 of r*, the centre of this third plane pencil lies also on /^. 



In a plane n through 2 of the points Hj there lie plane pencils 

 round both these points, hence also a third plane pencil; to this 

 belong the points of intersection of n with the |)erpendicular bisector 

 planes through the third of the points Hj^ hence: 



to r there belong three bilinear co?ig)'uences, lahich have as direc- 

 trices the join of 2 of the points Hj and the line through the S''^^ of 

 the points Hj and M. 



If jx passes through 2 points Hj and through M, the complex 

 rays in :rr for-m the plane pencils round these three points. 



In a plane .-r through one point Hj and two of the points Di 

 there lie three plane pencils of complex rays round these points. 

 If ri passes also through M it is a cardinal plane. 



^ 13. Before investigating the planes through a point Di I shall 

 first consider the complex cone of a point P of the perpendicular 

 Wi out of M to one of the side planes of T. This conq')lex cone is 

 apparently split up into three plane pencils, lying in the perpendi- 

 cular bisector planes through ini; mi is a threefold generatrix of the 

 complex cone of each of its points, hence: 



the four straight lines nii are 3 fold rays of P. 



In a plane ji through mi lies a plane pencil round Mand a plane 

 pencil round />,•; now the /^ of :t is a double ray and nii is a 



