441 



The sextuple involution, wiiicli [o"] deteroiines in a plane (p, has 

 llnwe slnijiiliir jioiiits S of order tiro Ijing in a straight line x and 

 (in the intersections of .i") four simjulnr points of order one, which 

 are completed into sets of six by the pairs of an involution lying on .?. 



Any frisecant t of a q" is trisecant of oo' curves of the congruence 

 and in particular of a figure ((/, ;?"). The congruence of the singnJar 

 trisecants is therefore identical with Ihe congruence of the chords 

 of {V, is consequently a (2, 6). 



The cone projecting a r»" out of one of its points has in common 

 with ö" the 6 intersections of the two curves; the remaining 9 points 

 determine each a singular bi.secant h. 



The surface W belonging to a point ^S' of o' consists of 2^, the 

 plane a (of which any straight line is singular bisecant) and a cone 

 {by. Consecpiently the simjuhir hisecants b form a congnience (9, 12). 



A plane if contains a curve 7' being the locus of the points of 

 contact of curves (>". As 7' has 34 points in common with ./*", 

 outside o', the curves tf touching 7 form a 't>'\ which is moreover 

 intersected by 7 in a curve '/'■". As '/" is intersected by an arbitrary 

 2^ in 10 points, o' is decuple curve of *'' ; so </" has three 

 octuple points *S'. From this it ensues further that </' and </'\ apart 

 from the points S, have 96 [joinls in common, so ihai </ is ogculaied 

 by 48 curves (j". 



As (/* has outside (f 140 points in common with V'^ there are 

 140 curves ()" touching two planen. 



The bilinear congruences of twisted curves (>^ and (^>\ which are 

 determined by nets of cubic surfaces 1 have considered in comnui- 

 nications published in volume XVII, p. 1250, in volume XVIII, 

 p. 43 and in vol. XVI, p. 733 ami 1186 of these Proceedings. The 

 congruence of twisted cubics determined by a ['!'"] was extensively 

 treated by Stuyvakut (Bull. Acad, de Belgique, 1907, p. 470 — 514). 



Mathematics. — "Associated points ivith respect to a complea; of 

 quadrics." By Chs. H. van Os. (Communicaied by Professor 

 Jan de Vries). 



(Communicatud in the meeting of May 29, 1915). 



Let a triply infinite linear system {comp/eu'^ be given of (piadrics 

 'ƒ>^ The surfaces passing through a point P form a net and have 

 moreover seven points Q in conunon. If we associate those points 

 to P we get a correspondence, which will be considered here. 



