590 
cally separated from P by the points A, and D,; it approaches 
therefore also to D, and in such a way that lim. PD: P'D, = —1. 
In the same way the point Q on the conic 4? approaches to the 
point D,. From this it is easily seen, that in the limit the line 
P'Q' coincides with PQ so that the line / is a ray of coincidence. 
Consequently the straight lines through these six points D are also 
rays of coincidence. 
A line ¢ is also a ray of coincidence, if P’ coincides with Q and Q' 
with P, so that the supporting points P and Q are associated to 
each other in the involution (P, P’). According to § 8 these rays 
form a cubic complex. 
§ 13. When a straight line ¢ describes a plane pencil, the associ- 
ated ray ¢' describes a ruled surface R, of which we shall determine 
the order. 
Each ray is bisecant of one curve 9°; the locus of the supporting 
points is a curve c; this has a node in the base point B of the plane 
pencil, for on the two rays ¢ connecting B with the two other points 
of intersection of the v* passing through £4 one of the two supporting 
points gets into B. Hence the curve c is of order four. 
The curve c* has one point in common with each of the three 
lines a,, a, and a,; for if a ray ¢ intersects one of these lines, one 
of the two supporting points gets into the point of intersection. 
Through the involution (P,P') a curve (/)* is associated to a line 
l, hence to a curve of order four, in general one of order twelve. 
The curve of has one point in common with each of the straight 
lines a,, a, and a, and to each of these points a line 2 is associated, 
so that moreover a curve 9° is associated to the curve 9°. 
The pairs of supporting points form on the curve c* an involution 
with sir coincidences; these are the points of contact of the six 
tangents which can be drawn from the node B at the curve c’*. 
The pairs of points of the curve 9’, associated to them, form there- 
fore also an involution with six coincidences. The lines connecting 
associated points of this involution form consequently a ruled surface 
of order siz, which is the surface R. 
We can also determine the order of & by trying to tind the 
number of points of intersection of this surface with the line a,. 
With a view to this we remark that to the line a, a surface w’ is 
associated, so that whenever one of the supporting points of a ray ¢ 
lies on this surface w?, one of the supporting points of the associated 
ray t’ lies on the line a,. The surface w? passes through the lines 
the curve cf intersects this surface besides in the points 
a, and a,; 
