593 
K*, project the lines d from the corresponding points Q’ of the 
t,". In the same way as in § 11 we should therefore find that these 
rays form a congruence (5,3). But it happens three times that the 
point Q’ coincides with the point D and hence lies on the line d; these 
points are associated to the three points of intersection of the curves 
o* and t°, for in them the point Q coincides with the point D’. In 
this case all rays through the point Q’ intersect the line d. Accord- 
ingly, from the congruence (5,3), which we should find in general, 
three sheaves are split off and we only find a congruence (2, 8). 
To the sheaf of rays through a point 5 of the line A,A, are 
associated one sheaf, two fields of rays, one bilinear congruence and 
one congruence (2,3). In general there corresponds therefore to a 
sheaf of rays a congruence (4, 6). 
§ 16. To a field of rays corresponds also a certain congruence. 
In order to investigate this, we consider the rays lying in a plane 
a through the line A, A, 
A non singular ray of this field is bisecant of a conic &? in this 
plane z, hence associated to another bisecant of this conic. To the 
congruence in question belongs therefore in the first place the field 
of rays itself. To the line A, A, in the plane z correspond a bilinear 
congruence of rays and two fields of rays. 
To an arbitrary straight line through the point A, corresponds a 
quadratic cone with the point A, as vertex. This intersects the plane 
a along two straight lines. The sheaf of the rays through the point 
A, belongs therefore also to the congruence in question and each of 
these rays must be counted twice, because it is associated to two 
rays of the plane a. The same holds good for the sheaf of the rays 
through the point A, 
The plane zr intersects the curve 0° besides in the points D', and 
D', in one more point; through this point passes a plane pencil of 
singular rays of the congruence (1,3). To each of these rays corres- 
ponds a plane pencil, which projects the line d, belonging to the 
conie £°, from a point of this conic; hence to the plane pencil 
mentioned corresponds the congruence of the lines resting on 4? and 
d. As these have a point D in common, those lines of intersection 
form a congruence (1,2). A field of rays is therefore transformed 
into a congruence (6, 6). 
