1253 
have their nodes on xs, consequently contain oo! curves 0°, inter- 
secting s in the same two points. 
The 0° of the monoid }** are represented on the plane y — F’, F, FP, 
by a pencil of g*, which have the intersection D of s as triple 
point and pass through /,, /,, /,. The remaining base-points /,, 
EK, E, of that pencil lie in the intersections of straight lines py 
of the monoid, which lines meet in S* and apparently are parabolic 
singular bisecants. The sixth straight line of the monoid passing 
through S* is the bisecant 6 of 6°, consequently part of a de- 
generate 0°. 
The straight line D/’, is the image of the conic @,*, in which 
the monoid is moreover intersected by the plane (s/’,); the nodal 
y* completing it into a y* represents the cubic 9°, belonging to @,’. 
So three figures (o’, 9’) lie on 2*. 
The straight line DE, forms with the nodal cubic passing through 
KE, E,, F,, F,, F, and twice through D, the image of a degenerate 
0°, consisting of the straight line h in the plane (sp,) and a rational 
o* passing through S*. The monoid 2* too contains therefore five 
figures (Db, 9%). 
5. We can now determine the order of the locus of the rational 
curves of. It has s as quadruple straight line and passes thrice 
through 6° ($ 3). Its intersection with a 2* consists apart from 
these multiple lines of five curves 9‘, is therefore of order 33. The 
rational curves 9* lie therefore on a surface of order eleven. 
The section of this surface ®'' with the plane (F's) consists of 
the quadruple straight line s, and parts of degenerate figures of. 
To it belong in the first place the three straight lines joining /’, to 
the intersections S,® of o* (§ 2); the remaining section is formed 
by the two @,’ belonging to the bisecants 6 out of the points 
\',C,' (§ 2). A straight line passing through /’, intersects ®'* four 
times on s and has with each of the two conies 0,’ a point of 
intersection not lying in /,; so five intersections lie in F. The 
three fundamental points F are therefore quintuple points of DP", 
In order to determine the locus of the intersection 5 of a o' 
with the bisecant coupled to it, we consider on s the correspond- 
ence between its intersections with 4 and eo“. Through any point 
P passes one 6; to it are associated the three points Q, which of 
has in common with s. In each point Q, s is intersected by four 
4 
curves v‘; hence four points P are associated to Q. From this it 
appears that s contains seven points 4. A plane passing through s 
contains three straight lines 4, consequently three points B; so the 
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