588 
has one point in common with the line a,. To this point corresponds 
a line 4, so that the curve (/)* which corresponds to the line /, 
must degenerate into this line 2 and into a conic 47, the locus of the 
pairs of points (E’, FF’). These pairs of points form an involution 
on the conic £7; the line e’#’’ passes therefore through a fixed point, 
so that to the line 2 a plane pencil of the plane «,, is associated. 
The same holds good for a straight line in one of the planes 
a 
Ong, U hon Gay GUNG! Ge 
25? 
According to § 4 each transversal g,, of the lines 6,,, and 6,,, 
contains an involution of pairs of points (G,H) which are each time 
the supporting points of a curve o°. The associated points G’ and A’ 
lie on the curve (/,, which through the involution (P, P’) is associated 
to the line g,,. The pairs of points (G’,H’) form an involution on 
this line with two coincidences and the lines G’H’ determine a _ 
quadratic ruled surface, associated to the singular line q,,. 
In the same way there corresponds to each of the lines g,, and 
Js, A quadratic ruled surface. 
The straight lines which are associated to all the lines g,,, form 
together a line complex, the order of which we shall determine 
later on. 
$ 11. It appeared in § 5 that on each degenerate ge’ of the first 
system lies one singular point D’ which is associated to all the points 
of the line d. A bisecant / of this o° through the point D’ corres- 
ponds therefore to a plane pencil which projects the line d from 
the point which is associated to the second supporting point of the 
bisecant. 
These bisecants 7 form two plane pencils, which both have the 
point D’ as base point; the first lies in the plane of the conic 4’, 
the second projects the line d from the point D’. 
The plane of the conic 4? passing through the line 4,A,, the 
bisecants / of the first kind are the common secants of the line A, A, 
and of the locus o* of the points D’. As A,A, and o* have two 
points D', and D', in common, their common secants form a con- 
gruence (1,3). 
A plane JV intersects the curve o* in three points; through each 
of these points passes one bisecant / of the second kind lying in the 
plane WV; these biseeants form consequently a congruence of class 
three. 
From a point P the curve 6? is projected by a cubic cone K°. 
The planes which project the corresponding lines d from P, envelop 
a quadratic cone of which the tangent planes are projectively asso- 
