65 



To each of the six straight lines p belongs therefore one ()'' that 

 has a pencil of parallel planes of symmetry, or: 



through 5 given points there pass six cylinders of revolution; their 

 generatrices are parallel to 6 sides of a quadratic cone. 



§ 13. The planes of symmetry through an arbitrary point touch 

 a cone of the S''^ class; let Jt be such a plane through the centre 

 M of the sphere B through 4 of the 5 given points; rr is then a 

 plane of symmetry of an 0^ thiough the 5 points and also of the 

 sphere B, hence of a pencil of (P's through those 4 points, or: 



througlt the centre of the sphere throngh 4 given points there pass 

 cc^ planes each of ivhich is a plane of symmetry of a pencil of 0^'s 

 through those 4 points; these planes envelop a cone of the 3''^ class. 



Such a plane .-r is also a plane of symmetry of the base curve 

 of the corresponding pencil, consequently to this pencil there belongs 

 a cylinder of revolution of which the generatrices are perpendicular 

 to jr, hence: 



through 4 points there pass oo^ cylinders of revolution of which the 

 generati'ices are parallel to the generatrices o/ a cone of the ^''^ order. 



\ 14. If six points are given, we consider two groups of five 

 points; these have 4 points in common. The surfaces of the 3"' class 

 corresponding to these two groups, have in conimon the tangent 

 planes of a developable surface of the 9^'' class that has V^ as a 

 4-fold tangent plane. However, we must subtract from this the 

 tangent planes through the centre of the sphere thiougii the 4 com- 

 mon points, hence : 



the planes of symmetry of the 0*'s throngh six given points envelop 

 a developable surface of the 6"' class that has V^ as a 4:-fold 

 tangent plane. 



§ 15. The quadratic surfaces through six points cut V.^ in a 

 (^•')5 ; to this there belong 4 double straight lines; how many degene- 

 rate curves touching y* twice, belong to (X;*),? 



In order to determine this number we remark that the cone in 

 R^ formed by the straight lines joining the image of y" to 0\ 

 (^ 11), cuts the image R^ of [k*)^ along a cur\e k* of the 4^^' 

 order; this curve has besides the 4 points that are the images of 

 the double lines of (Pjg and that are to be counted twice, 4 more 

 points in common with F\, hence: 



through six points there pass 4 parabolical cylinders and 4 p<ira- 

 boloids of revolution. 



