642 
pair of lines. Also here we find three systems, each consisting of a 
fixed line and a ray of a plane pencil. 
4. The line g rests on a,, a, and 8’; the line A cuts a, and 8? 
Through the point B of B? passes one transversal g,, =A, 4,; the 
corresponding point A, determines the plane of 2? and in this: way 
the point B’ of 6’; h, =A, B’ forms with g,, the pair of lines. We 
find therefore three systems of pairs of lines in group 4. 
Let us consider the correspondence (5,5’). Any ray h, of the 
plane pencil (B'A,) is eut by two rays g,, of the scroll corresponding 
to A,; the transversal through B’ of a, and a, is associated to a 
definite point of a,, and intersects the corresponding ray A, in B’. 
Hence the ruled surface of the pairs of lines g,, which we have 
associated to the rays h,, intersects the plane (B’a,) along a cubic 
passing through 4’. But in this plane lies a line g,, connecting the 
points A,,4, in (B'a,). The ruled surface (g,,) is therefore of order 
four; it intersects 8? besides in B’ in seven points B, which in the 
correspondence in question are associated to B’. Each of the eight 
coincidences is the double point D, of a pair of lines; the locus 
of D, is for this reason a twisted curve of order eight, d,°. 
The lines g,, form a ruled surface of order four with nodal lines 
dy, A, and directrix B. To each point A, are associated four points 
D,, while to a point D, there corresponds one point A,. From this 
follows, that the order of the ruled surface (,) with director lines 
a, and d,°, is twelve. 
3. ORDER AND cLass. With a view to defining the order of the 
congruence, we consider the conics 4? through a point P in 8. To 
them belong in the first place the three pairs of lines of group 1, 
each formed by one of the lines g,,, together with the line through 
P and the point (g,,,, 8). Further the six pairs of lines of group 2, 
defined by the three rays PAz. As each of these three rays belongs 
to two pairs, we come to the conclusion, that the order of [47] 
is nine. 
A plane through an arbitrary line & intersects a, and a, in the 
points A,,A,, and a, in a point A’,, which we associate to the 
point A, corresponding to A,,A,. Of the scroll (y,,) defined by A,, 
two lines rest on £; hence two points A’, are associated to A,. As 
A', coincides three times with A,, three planes A,A,4, pass through 
k, which is consequently a chord of three conics 47. The class of 
[27] is therefore three. 
4. SINGULAR CHORDS. According to a well known property of the 
