55 



^ 6. All the straight lines tlii'ongh (he eenire M of the sphere H 

 passing thmiigh the points Ai, aie rays of F. Likewise all the straight 

 lines perpendicnhir to a side plane of the tetrahedron T that has 

 Ai as angular points ; for they are axes of the C consisting of that 

 plane and a parallel plane throngh the 4''' angular |)oint. Further 

 any line perpendicular to 2 subtending sides of 7'belongsto T; they 

 are the axes of the (9" consisting of the pair of parallel planes 

 through these 2 sides, hence: 



the complex r has 8 cardinal poinis: M, the points /), at injivity 

 of the normals to the side planes and the points at infinitij Hj of 

 the normals to the subtending sides of T. 



^ 7. If the points Ai revolve round a straight line /, lying in 

 a perpendicular bisector plane of a side of T, 2 of the points Ai 

 describe the same circle; from this follows that / belongs to r, or: 



the six perpendicular bisector planes of the sides of T are cardinal 

 planes of P. 



I shall now show, (hat all the straight lines of V are double 



' O CO 



rays of P. 



^ 8. The axes corresponding to an arbitrary point Pof V^ belong 

 to a pencil {()'')'•, they are the straight lines // conjugated to the 

 polar line p of P i-elative (o y^ The centres of the individuals of 

 the pencil lie on (he polar line p of P relative to y' (they belong 

 to the parabolical cylinder of the pencil) and on a conic passing 

 through P and M and intersecting p. The axes through P form 

 consecpiently a plane pencil in V^ and a pencil (he plane of which 

 passes through M, hence: 



tlie complex F coiisists of oo* plane pencils of parallel rays tyiny 

 in the planes of the sheaf round M. 



From this there follows (hat F is invariant for any homotlietic 

 transformation relative to M; the complex cones corresponding to 

 the points of a straight line through M, have accordingly the same 

 curve at intinity. 



§ 9. All the straight lines of V^ belong to F, hence (he complex 

 curve of an arbitrary plane re touches the /^ of its plane. Besides 

 this straight line one more tangent may be drawn to the complex 

 curve out of each point P of this /^^, namely the line of intersection 

 of (ft with the plane of the pencil of complex rays through ^passing 

 through M. Consequently l^ is a bi-taugent of the complex curve 

 of Jt and also of all the planes in which it lies, or: 



V^ carries a field of double rays of P. 



