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In any arbitrary plane pencil of rays (/) a trilinear correspondence 
is similarly originated. Each of the three double-rays (coincidences) 
is a transversal of three corresponding rays b,c,d, and therefore a 
line ¢’ of a ruled surface (bed), which passes through A and is con- 
seqnently conjugated to a ray t4. 
Hence the rays which in the correspondence (¢, t') are conjugated 
to the rays through A form a cubic complex. 
Similarly every ray ¢, (in the plane a) is also conjugated to a 
regulus; so the rays of the plane a, denoted as a whole by [vl], too 
are singular. 
6. We now consider a ray ¢ of the sheaf of lines which has 
D* = apy for its vertex; let d be the ray of (D,d) which meets t. 
To ¢ now are conjugated all the rays ¢’ of the plane pencil which 
has D* for its centre and is situated in the plane D* d. Hence this 
pencil consists of singular rays which are each conjugated to every 
ray of the pencil. The sheaf [ D*] contains o’ of such singular 
pencils of rays, the planes of which pass through the line AA*. 
By an analogous reasoning it is found that the rays in the plane 
de = ABC are singular and constitute o* plane pencils conjugated 
to each other and having their centres on the line dd”. 
Hence the involution (é, 7’) contains eight sheaves and eight planes 
of singular rays. 
The ray AB meets two definite rays c,d, but al/ rays a, 6, and 
is therefore conjugated to all transversals of these rays c, d. Similarly 
the ray ag is conjugated to oo’ rays ¢’. Thus there are twelve principal 
rays, each of which is conjugated to all rays of a bilinear congruence. 
7. The hyperboloids (bed) which pass through <A, constitute 
evidently a double infinity. Thus through each ray a passes one of 
these (bed). Hence every ray a forms with three definite rays 
b,c,d a quadruplet belonging to one and the same hyperboloid. 
The transversals ¢ of this quadruplet form a regulus of which any 
two lines in the involution (¢, ¢’) are reciprocally conjugated. We 
shall call such a regulus singular. 
Thus the pencils (a), (4), (c), (d) can be made projective in such a way 
that every four corresponding rays are directrices of a singular regulus. 
We now consider the congruence which contains the oo! singular 
reguli. 
In any plane ¢ the projective pencils (v7), (6), (c) determine three 
projective point-ranges; hence in p there lie three lines ¢, each inter- 
secting four corresponding rays a, 6,c,d. Similarly any arbitrary 
point carries three lines of the congruence. Hence the singular regult 
form a congruence (3,3). 
