587 
responding points #,’ and #, describe apparently two projective 
sequences of points. Moreover the pairs of points (#,, F) form an 
involution on the line a,; the pairs of generatrices (2, u) form there- 
fore also an involution. Consequently the pairs of points (£,', F,') 
form an involution on the surface w? and the lines connecting 
associated points of this involution are the rays associated to the 
line a, 
We shall first demonstrate that each generatrix v of the surface 
w* which belongs to the same system with the lines a, and a,, 
contains one pair of points (#,', F,'). With a view to this we remark 
that two points #,’ and F,' are situated on the same curve 9°; 
Q 
this curve intersects the surface w° besides in the supporting points 
of the bisecants a, and a,. The congruence [g*] being bilinear, each 
line wv belongs as bisecant to one o°; the corresponding supporting 
points are the points in question /,’ and F,’. 
Through a point Z, of the surface w? there pass two rays of the 
congruence in question, viz. the line connecting Z,’ with its associ- 
ated point /,’, and the line v passing through the point £,’; con- 
sequently the order of this congruence is two. 
A tangent plane of the surface w* contains one line v and one 
line 2. The straight line u, associated to the line A, cuts this tangent 
plane in a point Ff,’ and the line connecting this point with the 
associated point #,’ is a ray of the congruence in consideration, 
which together with the line v lies in this tangent plane. For this 
reason the class of the congruence is two as well. 
Analogous considerations hold good for the lines a, and a,. Con- 
sequently to each of the lines a,, a, and a, theré corresponds a 
congruence (2,2). 
4 
§ 10. A straight line / through the point A, is bisecant of oo! 
curves o°. The point A, corresponds to itself; the locus of the points 
P’ corresponding to the points P of the line / is according to § 8 
a curve (/)*. This passes through the point A,; for when P gets 
into A,, P’ coincides with P. The rays associated to the line / project 
the curve (/)? from the point A, and form therefore a quadratic cone. 
The same holds good for a straight line through the point A,. 
A straight line / in the plane «,, is bisecant of oo! conics k?. Let 
E and F be the points of intersection of the line / with such a 
conic. The points £’ and /’, associated to these points / and F, 
lie according to § 5 also on the conic 4? and the straight line 2’ F’’ 
is associated to the line /. 
The locus of the points 4’ and /” is a conic k?, for the line / 
