Mathematics. — “Quadratic involutions among the rays of space.” 
By Prof. JAN De Veres. 
(Communicated at the meeting of December 28, 1918). 
In a communication which is to be found in part Vol. XXII, 
p. 478 of these Proceedings I have dealt with an involution, the 
pairs of which eonsist of the transversals to quadruplets of straight 
lines belonging one to each of four given arbitrary plane pencils 
of rays. In the sequel [ shall consider a few involutions related to 
the above mentioned. 
1. In the first place we assume two plane pencils of lines 
(A, a) = (a), (B, 8) = (6) and a quadrie regulus (c)? te. one set of 
generators of an hyperboloid I. An arbitrary line ¢ meets one ray 
a, one ray 6 and two rays c. If we conjugate to ¢ the second 
transversal # of these four lines, a quadratic involution among the 
rays of space is thereby detined. 
If ¢ describes a plane pencil, an involution is thereby determined 
in (c)*, the pairs of which correspond projectively to the rays of 
the pencils (a) and (6). 
Now consider the more general case where a quadratic involution 
in (c)? is brought into a projective correspondence to the pencils (a) 
and (6) in an arbitrary way. The transversals 7, ¢’ of the quadru- 
plets of rays a, 6,c,c’ will constitute a ruled surface, the order of 
which we shall determine by an investigation after the number of 
lines ¢ which rest on the line of intersection of the planes « and ~. 
On the line @f the projective pencils (a), (6) determine two 
projective point-ranges. Through each of the two united points (coin- 
cidences) passes a line £ The remaining rays ¢ which meet «ag, lie 
in « or in 8. 
On the intersection of I? and a the points of transit of the pairs 
c,c’ constitute an involution; the joins of the pairs of this involution 
form a pencil (Ce), which is projective to the point-range cut out 
on «-by the pencil (6) and therefore also projective to the line-pencil 
which projects this point-range from C. Since each of the two united- 
rays (coincidences), rests on four corresponding rays a, 6,c,c’ there 
are in « (and in 8 too) two rays of (4). Hence the ruled surface (t) 
is of degree six. 
The plane « intersects (¢?) still along an additional curve a‘, which 
