63 



§ 6. Tliroii^h consideration of the perpendicMilar bisector [)iane of 

 the straight line through the 2 points that belong to tlie gionp of 

 5 and not to tlie group of 4 points, we find that in this plane and 

 accordingly in each of the 15 perpendicular bisector planes, there 

 lie 2 generatrices of q\ 



The quadratic surfaces through six points cut F^ in a linear 

 system of oo' conies {^'')3. These define together with y* a linear 

 system (P)^ ; the tangents of the conic of the double lines of {k")^ 

 are the chords of contact of the 0^'s through the six points; polari- 

 sation of these straight lines relative to y' gives a conic k^ ; to (^')j 

 there belong four double lines, originating from parabolical cylinders 

 (cf. § 4), so that the locus of the axes has a conic k'' and 4 straight 

 lines in common with V^, hence: 



9" ?'* rational; it has a double curve of the order \0 ; the \^ perpen- 

 dicular bisector planes of tlie joins of the six points are bi-tangent 

 planes; V^ is a 4:- fold tangent plane. 



^ 7. In order to investigate the axes of the 0^'b through seven 

 points, we consider a group of 4 and a group of 6 of these points 

 that have 3 points iji common. We get in this way a complex /", 

 and a ruled sui-face (>" that have 18 straiglit lines in common. If we 

 subtract from them three times two straight lines lying in the 

 perpendicular bisector planes of the joins of the 3 common points, and 

 twice 4 straight lines in V^, we have 4 straight lines left, hence: 



through 7 points there pass 4 O^'s. 



^ 8. We can also arrive at this result in the following way. All 

 quadratic surfaces through 7 points cut F^ in a (/(;''),; in connection 

 with y* this gives a (/;'), with 4 double straight lines, consequently 

 in (^*), there are four individuals touching y' twice. These belong 

 to the surfaces of rotation through the 7 points. 



§ 9. A quadratic surface of revolution 0* has a pencil of planes 

 of symmetry passing through the axis of rotation and therefore 

 defined together with this axis, and further generally one more plane 

 of symmeti-y perpendicular to the axis. I shall investigate these latter 

 planes for O^'s through given points and I define as a plane of 

 symmetry of an 0* the polar plane of the point /'' at infinity of 

 the axis of rotation; if this polar plane is indefinite the planes 

 through the chord of contact p ot the LP are considered as planes 

 of symmetry. 



