465 
scroll (c)* in common; for any sheaf [|C] furnishes one ray for each of 
the two (1, 1). 
6. The aval complex with axis d is transformed by the trans- 
formation (tt) into a quadratic complex {t’}?; indeed, to the two 
rays of the scroll (#)* representing the plane pencil (¢’), correspond 
two rays of the image-complex lying in the plane pencil (¢’). 
As [d] singles out one ray out of each plane pencil of singular 
rays, (dj? contains the two fields of rays [a] and [8]. Two congru- 
ences {4} have besides those two congruences (0,1) one more con- 
gruence (4,2) in common; from this appears again that a bilinear 
congruence is transformed into a (4, 2). 
The image (3,1) of a sheaf [|M] has four rays in common with 
the image (1,1) of the field (u). One of them belongs to the scroll 
(c)? and is associated to any ray that the corresponding sheaf { C'} has in 
common with [J/| and [u]. Another coincides with c; for [JZ] and 
lu] send each one ray to c, and ca. 
The straight line through J/ and the point Cy, in u belongs to a 
plane pencil that is associated to a definite ray ..; as also contains 
a ray of this plane pencil, the image-congruences (3,1) and (1,1) 
have this ray (¢,)in common. Analogously they have a ray tin common. 
The images of two fields of rays |u| and [u*] have two rays in 
common. One of them is the image of the straight line uu*, the 
other is the line c; this is associated to the two transversals of c, _ 
and cg in w and in u*. 
The image (1,1) of the field [u| has six rays in common with 
the image (4, 2) of a bilinear congruence with directrices d,, d,. To 
them belongs the ray of the scroll (c)? associated to the sheaf of 
which the vertex lies in the point (c,). They have twice the line c 
in common, for two transversals of cz and ca rest also on d, and d,, 
while one straight line of u rests on c,, cg. The transversal through 
the point (wcs) to d,,d, belongs to a plane pencil which has also 
one ray in wu; to both of them corresponds the same line ¢,. Analo- 
-gously the image-congruences have a straight line és in common. 
The sixth common ray is the image of the transversal of d, and d, in u. 
