471 
6. The ten straight lines bj, = Br Bi are principal rays for the 
involution (¢, ¢’). For bp, is a bisecant of all the curves 6’, hence it 
is associated to all the bisecants through the point P’7/ which is 
homologous with the passage Py of buu. 
The sheaf | Py] is therefore associated to the principal ray bx. 
A plane pencil (7) contains ten rays ¢, each resting on astraight 
line }d,; and-on a conic Bom connected to it. The corresponding 
ray U’ rests also on dy. 
Further four rays t belong to the complex {é*, which in the 
involution (¢,¢') is associated to the sheaf [27 |. Consequently the image 
of the plane pencil (4 has quadruple points in Bz and in Bj, so 
that nine rays ¢’ rest on bj. A plane pencil is therefore transformed 
into a scroll of the ninth order. The plane pencil contains eight rays 
of the complex {t’}*; in @ lie therefore eight rays ¢, of the ruled 
surface (£)°. Besides « contains the straight line p’, homologous with 
the passage p of the plane rt, and a directrix of the ruled surface. 
7. A sheaf with vertex M is represented by a congruence of rays 
[¢’]. Let N be an arbitrary point, u the bisecant through AN to the 
curve 8? which cuts the straight line MP twice. The passage Q 
of w corresponds in a birational correspondence to the point P’ 
which through the homology is associated to P. 
When Q moves along a straight line q, so that u describes a 
plane pencil, the bisecants ¢ (§ 4) of the corresponding curves 8° 
form a complex {f}°. The complex-cone of M intersects a along a 
curve «? and the homologous curve a’® contains the points P’ asso- 
ciated to the points Q of q. The correspondence between Q and P’ 
is, therefore, of the fifth order; consequently Q coincides seven times 
with P’. Through MN pass therefore seven rays ¢t’ of the image of 
the sheaf [M |. 
The sheaf has with its image the ray MA and the rays of the plane 
pencil (M‚,a) in common; hence M is a singular point of the image. 
Let u be a plane intersecting « along the line p’. The curves ? 
which have the rays ¢’ of the plane pencil (P’‚u) as bisecants, have 
five of their bisecants ¢ in the plane (Mp) and these define on p 
five points Q, which may be associated to the point P. Inversely 
a point Q yields three rays ¢’ in wu, which are bisecants of the p* 
having MQ as a chord. To Q three points P’, consequently also 
three points P, are associated. Whenever a point Q coincides with 
a corresponding point P, the ray ¢’ associated to t= MQ, lies in 
u; the field-degree of the congruence [¢’] amounts therefore to eight. 
The image of a sheaf is accordingly a congruence (7,8). 
