443 



of the points Q, which are associated to the points P of ]'", this is 

 consequently a surface of order seven, 'P' . 



This order is also easily found from the number of intersections 

 with a q" of the complex ; the latter intersects V in 4 points P, 

 contains, therefore 28 points Q, associated to it. 



The joining lines of associated points apparently form a con- 

 gruence (7,3). 



§ 4. If tiie straight line / is one of the straight lines PQ, con- 

 sidered in ^ 1, a 0'^ of the complex will intersect the straight line 

 / in two associated points, consequently contain six points only, 

 which are associated to points of /. The locus of those points is 

 therefore a twisted cubic (j'. The curve (j" has been replaced here 

 by the figure composed of / and the q' counted twice. The latter 

 intersects / in two of the four points which / has in common 

 with A' ; the two others are the double points of the involution 

 lying on PQ. 



Let us bring through I'Q a plane T', in which J'Q stands there- 

 fore for the stiaight line ƒ/,. Tiiis plane intersects q' moreover in a 

 point R outside //, ; the joining lines of R with the two points 

 on ƒ/, associated to it, must be the straight lines y^ and ƒ/,. We 

 see therefore that the three intersections of </,, y^ and g, are 

 mutually associated and that each plane V contains one set of three 

 associated points. 



A Q^ of the complex passing through two associated points lying 

 on (/j, intersects «Ï»" further in the 6 points associated to them and in 

 the 14 points associated to its two other intersections with V. As 

 the total number of intersections must be 28, the 6 points mentioned 

 first are nodes of «P'. The three q' belonging to g^, g^ and y, are 

 therefore nodal curves of *'. 



A q" passing through the three intersections of (/,, </, and </, inter- 

 sects *' further in the 5 points associated to them and in the 7 points 

 associated to the fourth intersection of q^ and F. From this it easily 

 ensues that the five points mentioned are triple points of 'I>". 



^5. UP lies on L* one of the associated points coincides with 

 P. If R is one of the others the locus of R may be inquired into. 



A p^ of the complex intersects A^ in 16 points, contains therefore 

 the 16 6 = 96 points R associated to them; that loons is con- 

 sequently a surface of order 24, A'\ 



A^ and A" intersect in a curve of order 96; it will, however, 

 degenerate : 



29 



Proceedings Royal Acad. Amsterdam. Vol. XVIII. 



