( 762 ) 



puint, toiR'li any given line, touch any given |)lane. From the t'ollowing 

 will be evident that in our case these numbers are 3, 6, 3. 



All the surfaces of the net iV"^ with the eight base-points ()■„ passing 

 moreover througii any ninth point (>.,, form a pencil with () as 

 common centre and OX, (>Y, (>Z as common axes. ?]acii surface 

 of that pencil touching one of the eight base-planes «,- touches them 

 all, so it belongs to the series. A pencil of ipiadratic surfaces con- 

 taining three surfaces touching a given plane, we lind fi = 3. 



All surfaces of the net Nt with eight base-planes «,, touching 

 moreover an arbitrary ninth plane «,, form a tangential pencil with 

 as common centre and OX, Y, OZ as common axes. Each 

 surface of that tangential pencil passing throngh one of the eiglit 

 base-points Oi, contains all these, so it belongs to the series. So q = 3, 

 as three surfaces of a given tangential ' pencil pass through a given 

 point. 



The number of surfaces of the series touching an arbitrary line 

 of the plane A^A^A^ is three, because this line cuts the locus of the 

 points of contact (fig. 4) in three points. As the line is assumed in 

 a common tangential plane, each of those three cases counts twice; 

 so V = 6, as is immediately confirmed analytically. 



So the indicated series of quadratic surfaces is a series (3, 6, 3). 



Indeed, we also obtain a series with eight common points and 

 eight common tangential planes possessing the same characteristic 

 numbers (3, 6, 3) when starting from a common polar tetrahedron, 

 a point and a tangential plane not passing tlirough this point. 



7. We have two more points to consider with respect to our 

 oi'iginal problem. Firstly, we wish to point out how the case in 

 which C breaks up into four quadratic reguli is easily realised ; 

 secondly, we must show that all quadruplets of corresponding rays 

 have hyperboloidic position as soon as this is the case with four 

 of those quadruplets. 



When the original part of the locus O^ is a regulus W^ the pairs 

 of transversals of the (iuadru[)lets of corresponding rays are the pairs 

 of generators of this regulus arrayed in a (piadratic involution. If 

 such a quadratic involution of pairs of rays is cut by a plane situated 

 arbitrarily a quadratic involution of jtairs of points is generated on a 

 conic; this involution is as one knows characterized by the property, 

 that the connecting lines of the points completing each other to a pair 

 pass through the same point. To realize the above-mentioned case of 

 decomposition of the locus (>" we start from an arbiti'ary regulus 

 ()'', whose generators we regard as paii'ed otf in involution in a 



