472 
The sheaf contains a transversal of the straight lines bj, and Paas 
the congruence (7,8) has therefore in each of the ten planes 8u & 
plane pencil. These planes are accordingly singular for (7,8). 
The complex {4*, associated to the sheaf | Bj] has with [M] a 
cone (£)* in common; to this corresponds a cone (s;)‘; for to the 
intersection «* of a and the former cone, in the homology [ea] a 
curve «’* is associated and this curve contains the passages of the 
corresponding rays sj. Hence the congruence (7,8) has singular points 
of the fourth order in the jive points B. 
Through .W pass five singular rays s; accordingly (7,8) has five 
singular points of the second order in the plane «. 
The plane « is singular for the congruence (7,8), for the complex 
{t’}, conjugated to the field of rays [te], has a cone (¢’)* in common 
with [Af]. If a ray ¢ revolves round P, the bisecants u of the 
curves 6? which have the rays ¢, as chords, form a complex fu}®. 
Through MZ passes one bisecant u of the 4* corresponding to ¢,; its 
passage Q may be joined to P’ and the straight line P’Q—= q may 
be associated to the ray ¢’, homologous with ¢,. Inversely the plane 
(Mq) contains five chords wu, belonging to five different curves g*, 
each defining a ray ¢, through P, so that five rays ¢’, are associated 
to the ray gq. Through M pass therefore siv rays u, each correspond- 
ing in the involution (¢¢’) to a ray ¢t, of the plane pencil (Pa). 
Consequently «@ is a singular plane of the sixth order for the con- 
gruence (7,8). 
This congruence contains the ten rays by; for these correspond 
to the rays MP’). 
8. Now I shall consider the image of a jield of rays. The plane 
u contains ($ 7) eight rays ¢ associated to eight rays ¢’ through a 
point /. The image of the field of rays [u] has therefore the sheaf- 
degree eight. 
Let p be an arbitrary plane, P’ the intersection of p with the 
straight line p’ homologous to the straight line p — «u. The complex 
of the chords of the curves 8° which have each a ray of the plane 
pencil (P,u) as a bisecant, has five rays ¢’ in the plane pencil (P’,¢)- 
The plane p contains accordingly five rays of the image of [u]. 
Hence a field of rays is represented by a congruence (8, 5). 
The points Bx are singular for this (8,5). For the plane pencil 
(P, u) contains four rays of the complex {t¢'},‘; two of them coincide 
with p, the other two correspond to the ray BP’. The plane pencil 
(Bj, p') belongs therefore twice to (8, 5). 
The field [u] contains one ray of the field [,,,] and one ray of 
