479 
On the line e8 the pencils (a), (6), as soon as they have become 
projective, determine two projective point-ranges. One of the united 
points (coincidences) thereof is the point «grt; through the other 
passes a line ¢’. Besides this ray ¢’ the line « 8 also meets the three 
rays ¢’, which lie in @ and the three rays ¢’ in 8. 
By « the surface (¢’)’ is intersected along a curve «* witha triple 
point at A. Through every point of intersection of @* and r passes 
a line # which coincides with its conjugated line ¢. Hence the double- 
rays of the involution (t, t') constitute a line-complex of the fourth 
order. 
4. In order to make sure of this by another reasoning we con- 
sider the threefold infinity of ruled quadrics (Bed). Through three 
points, arbitrarily chosen on the lines eg, ay, ad, there passes one 
ruled surface (hed). Hence the system is linear (comp/ex) and inter- 
sects « along a complex of conics @?. The conies which touch a ray 
a at a point B, constitute a system of single infinity. The ruled 
surfaces of which they are the curves of transit, also constitute a 
single infinity; the base-curve o* is at & tangent to a. Every chord 
t of eg“ passing through F lies on a ruled surface (bed), which is at 
R tangent to the ray AR; the two transversals of the quadruplet 
a,b,c,d thus coincide in ¢. Hence the cubic cone which projects 
o* from R, consists of double-rays t= t?’. 
In addition to these the point A carries a plane pencil of double- 
rays, lying in the plane a. In order to understand this we observe 
that every line ¢ of a belongs to a pair of lines of the complex {a’}. 
Let S be the point of intersection of ¢ and the second line of this 
pair. Since AS is at S tangent to the corresponding hy perboloid 
(bed), t represents the two transversals of a ray-quadruplet a, b,c, d 
and is therefore a double-ray of the involution (t, ¢’). 
In this way we find again that the double-rays constitute a line- 
complex of order four. 
Simultaneously it has become evident that this complex has four 
principal planes a,3,7,0 and, by analogy, four principal points 
ag, CD. 
5. We now consider three rays 6,c,d meeting a line t4, which 
passes through A. Each line ¢ which intersects 6, c,d, meets in « 
a certain ray a and is therefore conjugated to ¢4. Hence every ray 
t4 corresponds to oo! rays t’; otherwise: the rays of the sheaf of 
lines |A\ are singular. 
By the rays /4 a trilinear correspondence is established between 
the pencils (b),(c),(d); in fact, two arbitrary rays b,c determine a 
transversal ¢4, which defines in its turn the corresponding ray d. 
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