Farthermore there are 3 6. positions of A for which O coincides 
with 4, leading to a point common to 6‘ and (0). 
From this we may still conclude that the planes in which the 
quadrangle 1234 admits one right angle envelop a surface of 
class 36. As to this we have to bear in mind that any plane through 
a trisecant of 6* having the vertex of the right angle on that trise- 
cant must be counted twice as tangential plane. 
‘Likewise we find that the planes for which the quadrangle 1234 . 
admits two equal adjacent sides envelop a surface of class 33. 
§ 14. Let us finally consider the locus of the centres My of the 
cireles circumscribed to the triangles /mn in the planes 4 through /. 
Each of the two trisecants cutting / furnishes again a point at 
infinity: each of the planes through a point of 64 at infinity deter- 
mines three points of I, and each of the tangential planes of 7» 
through / contains a fourfold point at infinity. So we find a curve 
uw, cutting 7 in 18 points, with the rank 42. 
If 7 is the bisecant 12, the points M, and M, generate a plane 
curve w* with the midpoint J/, of 12 as sixfold point; for the 
sphere with 12 as diameter determines on o* the vertices of six 
rectangular triangles with 12 as hypothenuse. As we can once more 
assign A, and M/, to the points 4 and 3, u" is like o* of genus 
zero. So its singular points are equivalent to 26 double points. So it 
must possess besides the double points on y?,, and the sixfold point 
M, still four double points more. These can only originate from 
coneyelie groups 1, 2,3, 4. So we conclude: the planes cutting o* in 
four concyclic points envelop a surface of class 4. *) 
So the curve u°* corresponding to an arbitrary line / has four 
fourfold points in the centres of the circles each of which 
contains a quadruple of points of 6*. 
As it cuts PF, in two fourfold points more, we get D= 36. 
By means of r— 42 and p—0 we find p=0, A= 174. 
If / is trisecant 123, each of the points W,, M,, M, describes 
a plane curve of order seven with a sixfold point. 
1) This is in accordance with the results obtained by Mr. M. SruyvaeERT in 
his inaugural dissertation (Etude de quelques surfaces algébriques engendrées 
par des courbes du second et du troisième ordre, Gand, 1912; see Chap. I, Sur 
les plans coupant un système de lignes en six points d'une conique). 
60 
Proceedings Royal Acad. Amsterdam. Vol. XV 
