444 



• 1-. in het loons of the poiiils P, foineiding with tinu of tlio jioints 



associated to them. L* and L'^^ tcuch eacli olhci' akiiig tliis curve. 



2. ill tlie locus of the points P, coinciding with one of their 



associated ones while two more of the other points associated to 



them coincide as well. 



§ (i. In order to find the first of these curves we investigate the 

 locus of the points R, associated to the points of the section c* of 

 V with AV 



A <i>' of the complex intersects c'' in 8 points, contains therefore 

 8 X Ö =3 48 points R, so that the locus of /? is a curve of order 

 24, ()". 



The curve (>" intersects V in 24 points, of which 2 lie on each 

 of the three straight lines ij, and these are associated to tiie inter- 

 sections oi g with the associated (>' ; there remain '18, which must 

 lie on c\ and in each of which the point P coinciding already 

 with Q coincides now moreover with R. 



The locus wanted is therefore a curve of order eighteen, p'^ 



\ 7. The o" found just now intersects L"^ in 96 points; 36 of 

 them are Ijing in the just found intersections with c^ tiie 60 remain- 

 ing ones lie on A% coincide consequently with one of the associated 

 ones while two others coincide on c\ We see therefore that the 

 second of the curves mentioned in § 5 is really of order 60. 



§ 8. Tiie <ï>' of the complex passing through a point P of L*, 

 have a common tangent t in P. As they form a net two more points 

 are necessary to determine one of them. 



We now take these points infinitely near P, and in such a way, 

 that they do not lie with t in one plane. The surface <P^ thus 

 determined has two ditferent tangent planes in P. must therefore 

 be a cone which lias P as vertex. A^ is therefore nothing but the 

 locus of the vertices of the cones of the com/)led\ 



§ 9. The involution /' considered here is a particular case of 

 an /" investigated by Prof. Jan de Vries'). Three arbitrary pencils 

 [<!>'') had been given there. Through a jioint P passes out of each of 

 them one 'I>^ , these 3 '/»' will intersect moreover in 7 points outside 

 P. If we associate these to /■" v.e get the /" meant. 



The ƒ' considered above is acquired by taking the 3 pencils as 

 belonging to oiie and the same complex; in that case the three 'P' 



'-) These Proceedings volume XXI, p. 43). 



