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same as the transversal b,,,. The curve (0,,,, A,D,, C,C,) belongs 
evidently only to the pencil of degenerate o° which contain the line 
bis As component; the curves (bs, Oss: D,C,) and (6,,,, Oos DC) 
belong each to fwo pencils of degenerate 0°. There are therefore six 
curves of the first and as many of the second kind. Hence together 
with the curve (b,, A,A,, 6,) the congruence [9°] contains thirteen @° 
which have degenerated into three straight lines. 
§ 4. Singular points and bisecants of the congruence. The points 
of the three lines a,, a, and a, are singular points of the congruence. 
Let us consider for instance a point A, of the line a,. The 
surface ’,, through A, intersects an arbitrary surface p’,, along a 
curve 0°, which passes through the point A,. Through A, passes 
therefore a pencil of curves 09’; all these curves pass also through 
the second point of intersection of the surface g’,, witb the line a,. 
Also the points of the transversals bij, are singular points; for 
each of these transversals is component of a pencil of degenerate 0°. 
The straight lines through the points A, and A, are singular 
bisecants; for through any point of such a straight line there passes 
one oe’ and as this passes also through the points A, and A,, it has 
that straight line as bisecant. 
In the second place the straight lines in the planes a, a, ds 
a> Cas @, brought through the points A, and A, and the lines 
a,, a, and a, are singular bisecants. For each of these planes contains 
a pencil of conics k°, each of which is a component of a degenerate 
oe’, and a straight line in such a plane is bisecant of all these conics. 
In the third place the generatrices g,, of the surfaces y,,, which 
eross the lines a, and a,, are singular bisecants of the congruence. 
Such a line g,, is intersected by the surfaces g’,, in the pairs of 
points of a quadratic involution and the two points of such a pair 
are every time the supporting points of a curve @°. As the surfaces 
_@’,, pass through the lines a,, a,, 6,,, and 6,,,, the lines g,, are 
the transversals of the lines 6,,, and 3,,,. 
In the same way the transversals g,, of the lines 6,,, and 0,,, 
and the transversals g,, of the lines 6,,, and 6,,, are singular bise- 
cants of the congruence. 
The singular bisecants form therefore two sheaves, six fields and 
three bilinear congruences. 
a 
§ 5. Pairs of points on a degenerate 9’. 
o’. We now pass on to the 
consideration of the involution (P, P') and examine first what becomes 
of this correspondence, if the points P and /” lie on a degenerate o@°. 
