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more points; any point P of these eight determines with D,,D,,...,D, 
a pencil of curves c, with threefold points in these nine points. 
Any other point Q of D,D, determines a curve c, of this pencil 
having ten points in common with that line and breaking up there- 
fore into that line and a curve c, with double points in D,, D,, P 
and threefold points in D,, ),,..., D,. So any- point of intersection 
P of D, D, and j,, is at the same time a node of a c,. forming 
with D,D, ac, with nine threefold points. At first this result may 
seem astonishing; for we can indicate eleven points on D,D, each 
of which forms with D,, D,,..., D, a set of nine threefold points 
of a c,, and of these eleven points we find back eight only. But 
the three other ones prove to determine a c, (and therefore a pencil 
of curves c,) excluded from the beginning. 
To prove this we consider the net [d| of curves c, determined 
by the six threefold points D,, D,,..., D, and the double points 
D,, D,; the curve of Jacosr of this [d] is of order twenty-one and, 
as it passes five times through D,, D,, it is cut by the line D,D, 
in 11 points more. So D,D, contains 11 points each of which is a 
node of a c, belonging to [d]. 
Now let us consider the curve c, passing through D,, D, and 
admitting D,, D,,..., D, as nodes; this completely determined curve 
cuts D, D, in three points #, PF, G more. Each of these points lies 
on the curve of Jacosi of [d], for c, forms with the curve c, of 
(8) passing through / a c, of [d], of which the point / is a node; 
likewise these two curves form with the line D, D, a curve c, of 
which D,, D,,..., D, and Z are threefold points. However E does 
not lie on j,,, for this c, can be considered as the combination of a c, 
of (3’) and a c, and this combination has been excluded beforehand ($ 5). 
But it is evident that 4, F, G do lie on the curve j, quoted in § 4. 
The eight remaining points of intersection of line D,D, and the 
curve of JacoBr of [d] do lie on j,,; so on each of the 28 lines 
D;D, can be indicated eight points of /,,. 
Moreover j,, is cut by the conic D,, D,,..., D, in eight more 
points. These lie at the same time on the curve of Jacosi of the 
net |e] of curves of order seven passing twice through D,,D,,...,D, 
and thrice through D,, D,, D,. This curve of Jacosr of order eighteen 
is cut by the conic D,, D,,..., D, in eleven more points; of these 
however once more three do not lie on j,, ie. the points common 
to this conic and the curve c, passing once through D,, D,,..., D, 
and twice through D,, D., D,,. 
So on each of the 56 conics D; Dz, DD, D, D, can be indicated 
eight points of ),,. 
