637 
each conjugated to all the other, hence of singular rays. Now a plane. 
6 is intersected at two points S by the conic @, which I? has in 
common with @; every plane tangent to £'* therefore contains two 
pencils (s). 
In any arbitrary plane lie two points S, and therefore two rays 
s; through an arbitrary point pass two planes o and consequently 
four rays s. Since a second system of singular rays is obtained by 
interchanging a and 5 in the foregoing reasoning the pencils of 
singular rays form two congruences (4,2). 
The vertices of the pencils (s) lie on the conics a? and g°, their 
planes envelop the hyperboloid J”’. 
5. In order to obtain another involution among the rays of space, 
we consider two reguli (c)? and (dj, of the hyperboloids I’? and A? 
respectively. Any two rays c,c’ determine in combination with any 
two rays d,d’ a pair of transversals (¢, t’) constituting one pair of 
the involution which will here be considered. 
Now suppose that on T° an involution (c,¢’) be given which in 
some way is projectively related to an involution (d, d’) assumed on A’. 
The transversals of the pairs d,d’ form a linear line-complex, 
for, in a plane 4 the points of transit D, D’ of these pairs determine 
an involution on the transit (conic) of A*, so that the joins of the 
point-couples D, D’ form a pencil. This complex contains two lines 
/ of the second regulus of I’. There are therefore two transversals 
of pairs d,d’ which meet all the rays c. In addition to these two 
an arbitrary ray c meets the two transversals of the pairs in (c)’ 
and (d)? which are determined by c. Hence the transversals ¢, ¢’ of 
the pairs c,c’ and d,d’ form a ruled surface of degree four, 
denoted by (#)*. 
Evidently (f° contains also two rays of the second set of genera- 
tors, (d)? of A’. 
6. Thus to the rays ¢ of a pencil (7) t) corresponds a ruled 
surface (t’)?, which contains two lines y and two lines d. This 
surface meets the intersection gf of I and A’ at 12 points, eight 
of which lie on the last mentioned four lines; the remaining four 
carry each one ray c and one ray d intersecting + at two points 
which are collinear with 7’. 
This statement may be corroborated as follows. Through each 
point of v* pass a line ce and a line d, Their points of transit, C 
and PD, through r determine two point-ranges related by a 2,2- 
correspondence on the curves of transit y* and Jd’ of I? and A. 
