638 
The lines 7’C and TD are therefore reciprocally conjugated in a 
correspondence (4,4). Of the 8 united-rays of this correspondence 
four pass through the points of intersection of 7? and d?; the 
remaining four each meet a pair c,d the point of intersection of 
which lies on ef and therefore carries a ray ¢’ conjugated to a ray ¢. 
In addition to the two lines y already mentioned the ruled sur- 
face (t’)* has a twisted quartic y* in common with £*. This curve 
intersects t at four points, which are two and two collinear with 
T (§ 2). It follows from this that the double rays of the imvolution 
(t, t’) form a quadratic complex. 
The single directrix of (¢’)* lies in +t, the double one passes 
through 7. 
7. The rays of the reguli (y)’ and (0)? are evidently (§ 4) princi- 
pal rays of (t,t’). To each of these rays a bilinear line-congruence 
is conjugated having two lines c or two lines d for directrices. As 
each line c acts as directrix to two congruences (1,1), there emanate 
two pencils (¢’) from each of its points. The congruences (1,1) cor- 
responding to the principal rays therefore constitute two quadratic 
complexes. 
In a similar way as in $ 4 we find a congruence of singular 
rays. Of the intersection e* of the byperboloids I and A?’ each 
point is the vertex of a pencil (S, 0) consisting of rays s which are 
each conjugated to all the others, hence singular. For, in fact, the 
plane o through the lines y and 0, which are concurrent at $, 
intersects o* still in the additional points C of y, D of 0 and Z. 
Evidently CH belongs to (c)’, DE to (d)*. Each ray of (S,o) meets 
two rays c’,d' at S and intersects the lines c= CEH and d= DE; 
therefore (S,o) consists of reciprocally conjugated rays s of the 
involution (¢, t’). 
Since the vertices of the pencils S lie on of and the planes o 
envelop a developable of the fourth class, the pencils of singular 
rays form a congruence (4,4). 
8. Any three rays c of a cubic regulus (c)* determine in combi- 
nation with each ray a of a pencil (A,@) two transversals, which 
form a pair of an involution of rays in space. 
By the rays of a pencil (¢) the rays of (c)° are ordered in an /?, 
the sets of which are projectively correlated to the rays a. To begin 
with we again suppose that this correspondence is established in an 
arbitrary manner; then the transversals ¢, ¢’ of the quadruplets of 
rays constitute a ruled surface which will here be investigated. 
