1269 
points of g lie on a surface of order twelve, passing seven times 
through g and twice through each of the base curves. 
If the point G of g lies on a’, the surfaces a? admit in G a 
common tangential plane, the plane through g and the tangent ¢ in 
G to a*; so these surfaces determine on the curve (b°c®) touching ¢ 
in G an /’ of associated points. The cone 4” projecting a’ out of 
G cuts any curve (b°c*) through G in a triplet of associated points; 
therefore these points lie on the intersection of 4? with the monoid 
x* containing all these curves. So, for any of the six points common 
to g and a base curve, (()’ breaks up into a twisted cubic and a 
twisted sextic. 
Any common transversal d of g, a’, 8° and y° forms with g the 
partial intersection of three surfaces a’, 6°, c° with two more points 
in common; these two points form a group of the /* with any pair 
of points of g. 
The transversals of g, a’, and p* generate a scroll of order six 
with g as fivefold line; for the cubic cones projecting «’ and §* out 
of any point G of g admit g as double edge and intersect each 
other in five lines of this seroll. On g this scroll has 10 points in 
common with y°, so it cuts y° outside g in 8 points. So, the base 
lines g, a, 6°, y° admit eight common transversals and therefore eight 
pairs of points belonging to w* groups of the L*. 
Evidently the eight lines d lie in the surface A* of the coinci- 
dencies; of this surface g is a fivefold line. 
$ 10. The pencils (a’), (6?) determine a bilinear congruence of 
twisted cubics 9°. In general any ray m of a pencil (M, u) is bisecant 
of one v°; the locus of the points Q, Q’ common to m and this o° 
is a curve (Q)* with a double point in M. In the manner of § 2 
we introduce as auxiliary curve the locus of the points 2, A’ still 
common to m and the surfaces c° through Q and Q’. The surface 
c? through M cuts (Q)* in M and in six points Q; so M is a six- 
fold point of the curve (f) and this curve is of order eight. 
The polar curve of M with respect to the pencil of intersection 
of (c°) and u intersects (Q)* in M and 4 x 3—2 = 10 other points, 
lying also on (f)*. So 4 x 8—2 x 6—10 == 10 points are arranged 
in associated pairs. So, the pairs of points of the involution I* le 
on the rays of a complex of order five. 
Any point G of g is associated to two points of g, the points 
common to g and to the curve (G)° corresponding to G. So g is a 
singular line of the /*; the pairs of points lying on it generate an 
involutory (2,2). 
83 
Proceedings Royal Acad. Amsterdam. Vol. XV. 
