591 
it has in common with the lines a, and a,, in sie more points, so 
that the plane pencil in consideration contains sir rays ¢ of which 
one of the supporting points lies on the surface w*; consequently 
there are sir rays ?¢’ intersecting the line a, 
In the third place we can determine the order of A by trying 
to find the number of intersections with the line 4,A,. For this 
purpose we remark that a ray ¢’ intersecting the line A,A,, if it is 
not a singular ray, must be bisecant of a conic £*. The two sup- 
porting points are associated to two points of the same conic, so 
that also the associated ray ¢ intersects the line A,4,. The plane 
pencil contains one ray ¢ intersecting the line A,A,; the associated 
ray ¢’ rests also on the line A,A,. 
According to § 11 there is a complex of order five consisting of 
rays ¢ associated to singular rays # which form a congruence (1,3) 
and each of which intersects the line A,A,. The plane pencil contains 
5 rays of this complex, hence the surface R five rays ¢’ of the (1,3). 
In all the line A,A, is intersected by six rays ¢’, so that the 
surface FR is of order sie. 
§ 14. We can now also determine the order of the line complex 
associated to the congruence of the singular rays g,, found in § 10. 
A singular ray 7’, intersecting the line 6,,,, is bisecant of a 
degenerate 0° consisting of the line 6,,, and a conic £° of the plane 
@,,, passing through the point of intersection of this plane with the 
line ¢’. The supporting points of the associated ray ¢ lie also on tbe 
line 6,,, and on the conic £°. Now the plane pencil considered in 
the preceding § contains one ray ¢, which intersects the line 0,,,; 
hence there is one ray ?¢’, which intersects the line 0,,,. 
The other five generatrices of the ruled surface f° intersecting 
the line 6,,, must be singular rays, therefore lines g,,. The plane 
pencil contains five rays associated to rays g,,; consequently these 
rays form a complex of order five. 
§ 15. To a sheaf of rays corresponds a congruence [!]. In order 
to determine order and class of this congruence [t’], we take the 
base point B of the sheaf on the line A, A, 
It has been found already in § 13 that to a ray ¢ intersecting 
the line A,A, a ray ¢’ is associated also intersecting A,A,. We shall 
now show that the rays ¢ and ?¢’ intersect the line A,A, in the same 
point. 
Let /* be the conie which has the line ¢ as bisecant, P and Q 
the corresponding supporting points, P’ and Q’ the points associated 
39* 
