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corresponding to the rays ¢4 and therefore passing through A then 
constitute a met (twofold infinite system) all the specimens where- 
of have the lines d,e and the transversal 4, through A of d and e 
in common. Through a point P therefore passes a single infinity of 
hyperboloids and these still have the transversal through P of d 
and e in common. Hence the lines ¢’ through P which are conjugated 
to the rays of the sheaf [A] form a pencil in the plane (P4,). 
10. There are still other singular rays. Each plane « through e 
contains two lines c. In « lies a pencil of rays ¢, which has the 
point of intersection # of e and ae for its vertex; these rays are 
singular, since they rest at # on a third line c and are therefore 
all conjugated to each other. 
The sheaf |E]| is therefore composed of pencils of singular rays. 
The plane d passing through d and aq contains a linec,; through 
each point D of d pass two lines c, hence oo! lines ¢, which rest 
at the same time on c, and ag. It follows from this that the plane 
of rays |d] is composed of op* pencils of singular rays. These have 
their vertices on the line d. 
11. Lastly we consider a ruled surface I'* with a double curve 
o*. The linear complex which can be laid through five generators 
ec of I* contains all the lines c. The four rays c which rest on a 
line ¢ meet besides the line ¢’, which by the complex is conjugated 
to ¢. The involution (4,4) then consists of the pairs of conjugated 
directrices of a linear complex; its double rays are the rays of this 
complex. 
Another well-known involution (¢,¢’) is originated by the pairs of 
reciprocal polar lines of a hyperboloid. Its double rays are the two 
sets of generators of the hyperboloid. 
