490 
the locus of these generators is a cubic cone, so that the deduction 
of the order of the complex of double rays, as enunciated by Prof. 
pei Vries in § + of his communication, holds good. 
3. The number of surfaces (bed) which pass through a straight 
line a, is evidently equal to the number of those which pass through 
three points, te. to five. If, however, only those ruled surfaces are 
cousidered on which @ belongs to the same regulus as 6, c and d, then 
we must exclude the regulus determined by the lines 6, c and d 
which pass through the points of intersection of a with 3, y and ) 
resp. Furthermore, for instance, the plane (a D) and the plane 
passing through BC and through the point of intersection of (a D) 
and py constitute a degenerated hyperboloid of our system, which 
passes through a. Of this kind two more specimens can be pointed 
out. Hence through a there passes only one non-degenerated ruled 
quadiic (bed), on which a,b,e and d belong to the same regulus. 
The deduction of the congruence (3,3) of the singular reguli, enunciated 
in $7 of the above-mentioned communication therefore remains valid. 
4. Now consider a ray ¢ which meets the lines AB and yd. 
The corresponding rays a and 6 lie in the plane p passing through 
AB and t and intersect on the line «8; the lines c and d determined 
by ¢ pass through the point of intersection of p and yd. Hence the 
conjugated lines ¢’ form a pencil with its vertex on yd, in a plane 
passing through AB. 
Hence the involution (t,t’) contains six additional bilinear congruences 
of singular rays. 
We now consider the line AA*. The corresponding lines 6; c and 
d pass through A*; a is indefinite. To the line t= A A* are therefore 
conjugated all the rays of the sheaf A*. Similarly the whole system 
of rays of the plane a* as a whole is conjugated to the line (—aa*. 
Hence there are eight additional principal rays, four of which are 
conjugated to the rays of a sheaf, the remaining four to those of a_ 
plane. Thus the total number of principal rays is twenty (vide § 6 
of Prof. pe Vries’ communication). 
5. To the rays ¢ of a sheaf S are conjugated in the involution 
(t,t’) the lines of a congruence 2. We shall now discover the number 
of lines of S which pass through an arbitrary point P. To a line 
u passing through P we conjugate the line ¢ which passes through 
S and meets the same rays a and 6 as u. If u describes a plane 
pencil with vertex P, then ¢ engenders a quadrie cone. If, further- 
more, we associate to a line ¢ the ray wu’ which passes through P 
and meets the same lines ec and d, then the correspondence (£,4/) 
too is quadratic. The correspondence (w,w') therefore is of the fourth 
