59 



^21. If in rr thei'e lies a |)laiie pencil with centre /-* at finite 

 distance, there are also 2 plane pencils with their centres on the 

 /^ of nr ; if rr does not pass through one of the cardinal points at 

 infinitj, the planes of these latter pencils pass through M, hence: 



only the planes tkrouyh the 8 cardinal points contain degenerate 

 complex carves (of. §^ 12, 13 and J 4). 



§ 22. As the null system [2,1] and the involution [2] are 

 invariant for central projection, we can construct the complex cone 

 of an arbitrary point P in the following way : 



We determine the points of intersection i)/ of the perpendiculars 

 out of P to the side planes of T with an arhiti-ary image plane t 

 and also the intersection of I'M with r. Then we construct the 

 double points of the quadratic involution in which the conies of the 

 pencil through Di cut an aihitrary straight line / through 0; we 

 fix this involution by means of the points of intersection of / with 



2 degenerate conies of this pencil. The straight lines joining P to 

 the double points in question, are generatrices of the complex cone 

 of P. 



§ 23. If the points Ai are coplanar, their plane « cuts an 0" of 

 the system in consideration along a conic k^ through Ai or is a 

 part of the 0*. In the first case the axis of 0^ lies in one of the 

 planes through the axes of symmetry of k* perpendicular to «; in 

 the second case the axis of CP is a straight line perpendicular to (t. 

 The axes of symmetry of the conies through Ai are tangents to a 

 curve of the 3'"^ class touching the line /^^ of a twice; the planes 

 through these axes and _l « touch a cylinder of the 3"^ class with 

 V^ as double tangent plane. 



The rays of r in an arbitrary plane .t touch also in this case at 

 a curve of the 3"^ class that has the l^ of its plane as a bi tangent. 

 The complex cone of an arbitrary point P, however, splits up into 



3 plane pencils the planes of which touch at the cylinder in question ; 

 a perpendicular to « is a triple generatrix of the complex cone of 

 each of its points, hence : 



if the 4 points Ai are coplanar, F consists of the tangents to a 

 cylinder of the 3''^^ class; V^ is the bearer of a field of double 

 rays; the vertex of the cylinder at infinity is the bearer of a sheaf 

 of triple rays. 



§ 24. Now consider the case that 3 of the points Ai lie on a 

 straight line a; then the 0''s must pass through a fixed point A 



