442 



§ 1. We first prove the proposition : Any straight line / joining 

 two associated points P and Q contains an involution of pairs 

 of associated points. Any pencil of tiie complex has one *' in com- 

 mon with the net determined by 1' and Q, and intersects / there- 

 fore along an involution containing the pair of points P, Q. If two 

 pencils have 07ie «P' in common (if they "intersect" as we shall say 

 for the sake of brevity) the associated involutions have moreover 

 one pair of points in common and so coincide. If the two pencils 

 do not intersect a third may be introduced intersecting each of them 

 and it may be seen that the involutions coincide in that case too. 

 All pencils therefore intersect I along the same involution, any pair 

 of points of it consequently determines an infinite number of pencils, 

 sets apart a net out of the complex, by which Ihe proposition has 

 been proved. 



§ 2. Let us determine the locus of the points P coinciding with 

 one of tlieir associated points. For this purpose we determine the 

 number of those points lying on the section q" of two *' of the 

 complex. The sets of eight associated points on q* are cut out 

 on Q* by the '/>" of a pencil (*') from the complex. Now a pencil 

 («f»*) contains suteen (*^), touching a twisted quartic of the first 

 kind ; this is easily seen by making the curve to degenerate into a 

 quadrilateral, each of the sides of which touches then at two *', 

 while through each angle passes one <7*', which must be counted 

 twice.') The number of points lying on it* amounts therefore to 16, 

 their locus is therefore a surface of order four, h". 



§ 3. What is the locus of the points Q, if P describes a straight 

 line /?. 



Any <P^ of the complex intersects / in two points P, and so con- 

 tains also the 14 poijits Q associated to them ; the locus of these 

 points is therefore a curve of order seven, o'. It has in common 

 with / the four intersections of / and A\ 



A plane V passing through / intersects q' outside / moreover in 

 3 points Q, each associated to a point P of /. The 3 joining lines 

 PQ, which we shall indicate by r/,, (/^ and (/, contain each an 

 involution of associated points. 



The locus of the points P of T", for which one of the associated 

 points Q lies in V consists of these straight lines and of the section 

 c" of V with A*. Now this locus is the section of F with the surface 



') Vide Zbuthen, Lehrbuch der abzdhlenden Methoden der Geometrie, 

 Teubncr 1914. 



