920 
locus under discussion. So it is of order siz. As it contains at the 
same time the lines touching 9* in Sp, these points are nodal points. 
To the two points of r,° lying in JZ correspon! two points Q 
coinciding with 7%, whilst to the point of M lying on 7,7) two 
points on 7), 7, correspond. From this ensues that the four points 
T have to be also nodes of ZI. 
So to a plane corresponds a surface of order sie with eight nodes 
and ten lines. 
§ 7. Let us now consider the correspondence between two points 
P,Q separated harmonically by a twisted quartic curve of the second 
kind of. As P bears three chords of 6*, it is conjugated to three 
points Q. To the points P of a line / correspond the points Q of 
a twisted curve 4°; for each plane through / contains six points Q. 
The three points Q corresponding to P lie in the polar plane of 
P with respect to the quadratic surface H° through o*. The plane 
II rotates about the polar line / of /, if P moves along U. So I’ 
is a trisecant of A’. 
The scroll of the chords of óf cutting / is of order nine; so nine 
of these chords also intersect /’. To these nine belong the two 
trisecants of of cutting /, each of which represents three chords; 
they have to meet /’, as they lie on the hyperboloid H* and are at 
the same time trisecants of A*°. The remaining three chords cutting 
land / determine the three points Q on /. 
§ 8. Each of the six tangential planes of of passing through / 
contains a point and two tangents of 4°; so this curve is of rank 
twelve and rests in sir points on óf. By SS, we represent the points 
of 6° lying in a plane drawn through /; the chord 6=S,S, is 
paired to the chord 6’=S,S, and now we consider the corre- 
spondence between the points P and P’ in which 5 and 6’ intersect 
1. As P bears three chords we find a (3,3). If 6 and 4’ intersect / 
in the same point P, only the third chord through P furnishes a 
point P’ not coinciding with P; from tbis ensues that the coinci- 
dencies of the (3,3) coincide by two in a double coincidency. So 
through / three planes pass for which b and 0’ intersect in /; the 
line 4 separating / harmonically from 6 and 6’ then contains four 
out of the six points Q, the remaining two lying on 6 and 6’. 
So the curve A° admits three quadrisecants. 
§ 9. Let / be a chord of o* and S, and S, the points it has in 
common with of. Through any point P of l pass two more chords 
