1147 
Mathematies. — “On twisted quarties passing through eight associated 
points’. By Dr. Cus. H. van Os. (Communicated by Prof. 
JAN DE VRIES). 
(Communicated in the meeting of December 18, 1915.) 
Let be given eight associated points Bj (4; = 1, 2,...8), the base- 
points of a net {®*| of quadries ®°. Through these points pass oo’ 
twisted quartics e*, the base-curves of the pencils (®°) in this net. 
Of this system some properties will be investigated, which are 
analogous to the properties of the pencils of plane cubics, investigated 
by Prof. Dr. Jan DE Vries '). 
$ 1. A plane V intersects a peneil (®°) of the net along a pencil 
(y*) of conics; the base-points of the pencil (4°) are the intersections 
of the plane V with the base-curve @° of the pencil (®*). Let us 
now choose for the plane V a stationary plane, i.e. the osculating 
plane a in a point of injlewion [ of the curve e*. The four inter- 
sections mentioned coincide now: the conics of the pencil (p°) have 
a contact of the third order in the point /. Consequently the twice- 
counted tangent ¢ in the point / also belongs to this pencil. One of the 
surfaces of the pencil (®*) is being intersected by the plane a along 
two coinciding straight lines; this surface must therefore be a cone, 
having the plane a as tangent plane. The stationary planes of the 
curves 9° are therefore the tangentplanes of the cones of the net | B*|; 
the tangents in the points of inylerion are the associated generatrices. 
These tangents ¢ form a congruence (4, 12). For an arbitrary 
point / determines a pencil (®*); as tbe latter contains four cones, 
four generatrices ¢ pass through the point P. And the vertices of 
the cones of the net [y?| form a twisted curve g°, of the sixth 
order; a plane V consequently contains the vertices of siv cones of 
the net, which are each intersected along fwo generatrices. 
Each of the planes a will osculate fwo curves v* in points of 
inflexion /. For such a plane intersects the net [#*] along a net 
[y*| of conics, which net contains the straight line ¢ counted twice. 
A point P of ¢ determines in the net {[y*] a pencil (y*), of which 
all figures touch in the point P, and in a second point P’ of the 
straight line ¢ The pairs of points (/,P’) evidently form an invo- 
lution /?; the two double points of /* are the points of inflexion 
wanted, as in each of them the base-points of a pencil (#*) coincide. 
If the straight line ¢ is drawn through one of the base-points B, 
1) These Proceedings Vol. XVII, p. 102. 
