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sects the latter moreover in a point P and the plane ¢ passing 
through F,, /,,/, in a point P’, which we shall consider as an 
image of P. As one 9° passes through any point P, the curves of 
the congruence lying on 2? are represented by a pencil of rational 
curves g*. Every g* has in common with the intersection gy* of 2° 
the five points, in which the corresponding 9° intersects the plane ¢ ; 
the remaining seven intersections of g* with pf are base-points of 
the pencil (y‘). To them belong the points /, /,, F,; the remaining 
four are intersections of four straight lines lying on &*. One of them 
is intersected by every 9° in S and in a point P, is therefore a 
singular bisecant p of the congruence ; the involution which the oo! 
curves 0° determine on it, is parabolic; so we might call p a 
parabolic bisecant. The remaining three straight lines d,, d,, d, passing 
through S are common trisevants of the curves 9°; on these singular 
trisecants as well the involution of the points of support is special, 
for each group contains the point S. 
The monoid * contains moreover two straight lines passing 
through S viz. the two bisecants of 65° cutting s, being consequently 
component parts of two o° degenerated into a straight line 4 and a 9’. 
The pencil (v*) has three double base-points D,, D,, D, and four 
single base-points ZF, F,, /,; it contains six compound figures : 
three figures consisting of a nodal ¢* and a straight line and three 
pairs of conics. 
Let us now first consider the figure formed by the straight line 
D,D, and the g?, which has a nodal point in D, and passes 
through the remaining six base-points. It is the image of a figure 
consisting of a bisecant 4 and a rational curve 9‘; for the plane 
passing through d, and d, has only one straight line in common 
with 2? so that D, D, cannot be the image of a conic passing 
through S. Consequently there lie on X° three straight lines 6 not 
passing through S, and therefore three curves ¢* passing through S. 
The conic passing through D,, D,, D,, ZE, F, is the image of the 
conic 9? which the plane (fs) kas in common with 2’: the conic 
to be associated to her passing through D,, D,, D,, /,, /; is the 
image of the ©? forming with g? a curve of the congruence [9°]. 
There are apparently there figures (o°, 09°) on =’, 
4. The curves 0°, meeting s in a point S* lie on the nodal 
surface °, which has S* as node. The monoids =** belonging to 
two points of s, have one g° in common; so the groups of four 
points which the 9° have in common with s form a /,*. There are conse- 
quently six e*® which osculate s, and three binodal surfaces ®* which 
