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Analogously the rays of the axial complex |c.| are associated to 
the singular rays tz; to each ray f2 correspond the rays ¢’ of a plane 
pencil belonging to |cz\. 
The intersection of the complexes |cal and |cg| is a bilinear con- 
gruence of which the rays are associated to the ray t=c. The 
straight line c is therefore a principal ray; indeed, we can consider 
two arbitrary points of c as passages P and Q. 
All the rays ¢ through a point P= Q=C of c are associated to 
the ray ¢’ joining P’ Q’; hence also ¢’ is a principal ray. When C 
moves along c, P’ and Q’ describe two projective ranges of points 
on c, and ca; P’Q’ describes a scroll (c)’.. The quadratic scroll (c)’ 
consists therefore of principal rays, each of which is associated to 
the rays of a star [C']. 
3. When ¢, revolves round a point 7, Cz moves along cs and 
the plane pencil with Cs as vertex of which the rays ?¢’ ent the 
line ¢', in [a] associated to tf, defines a congruence. The range of 
points which C, describes on cg, is projective to the plane pencil 
T') described by f,; when it is projected out of any point M on 
a, there will be two rays f„ which pass through the projection of 
the corresponding point Cz. Through M pass therefore two rays of 
the congruence. Any plane u contains one point Cs and also the 
passage of the corresponding ray ¢., hence one ray ¢’ of the con- 
gruence. The plane pencil (tz) is accordingly represented by a con- 
gruence (2,1). . 
As the ray 7” Cs in each of its positions belongs to the (2,1), 
(T" Cà) is one of the singular planes of the congruence. Also « is 
a singular plane, for it contains the plane pencil the vertex of which 
lies in the point of intersection C=C of c and 6. 
4. lf t describes a plane pencil (7’,t) in the plane t, its passages 
P and Q describe projective ranges on the straight lines p = er and 
q=8r. But then also the ranges of points which the homologous 
points P’ and Q’ describe on p’ and q’, are projective, so that P’ Q’ 
describes a quadratic scroll. Accordingly in the transformation (1, ¢’) 
the image of a plane pencil is in general a quadratic scroll. 
If ¢ describes a field of rays uw, the passages P and Q remain on 
the straight lines p=au and g= yu; P’ and Q’ lie in this case 
on the homologous straight lines p’ and q’. The jield of rays is 
therefore represented by a bilinear congruence. 
The ray ?¢’ in uw joins the points pp’ and qq’; it is therefore a 
double ray of the involution. 
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