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the general case of an arbitrary v. But we wish to give just now one 
theorem where o has already any arbitrary value: 
“If D,, D,,.... D, are g-fold points of a non degenerated curve 
C3, of order 30, these points are at the same time the base points of 
a pencil of curves of order 30, each of which passes @ times through 
OD), , ets. on: elton ss 
For the proof it will suffice to remark, that the nine points lie on 
a cubie curve w,; so the pencil mentioned is represented by 
C32 + hu,? == 0. 
§ 4. By D,, D,,...., D, we will henceforth denote arbitrarily 
chosen points; we represent by (8’) the pencil of curves c, passing 
through them, by B, the ninth base point of this pencil. So the 
principal resuits, obtained for o = 2, are the following : 
I. “The locus of the point forming with D,, D,,...,,D, a set 
of nine nodes of a non degenerated *) c, is a curve j, of order nine 
passing three times through D,, D,,...., D,”. 
I]. “This curve j, is also the locus of the points corresponding 
with B, in tangential point on the curves of pencil (@')’. 
Ill. “Let vw, be any cubic of (3) and c, any sextic passing three 
times through D,, D,,...., D,. Then the line joining the last two 
points common to uw, and c, will meet w, for the third time in the 
tangential point 7’ of B, on w,”. 
Before continuing our considerations we wish to correct the pre- 
ceding communication. We have indicated there that 2, does not 
lie on 7,; indeed this is so, but one of the proofs — the geometri- 
cal one — may give rise to difficulties. Therefore we once more 
prove here: B, does not lie on j,. To that end we consider j, as 
the locus of the points on any curve of (3’) corresponding with B, 
in tangential point. Now B, will be a point of j,, if and only if 
one of these points coincides with B, which only can happen if B, 
is a node for one of the curves of (9). Of these nodes — the 12 
so called ‘critical points” of the pencil — none however coincides 
with one of the base points, if — as it is the case here — eight 
of the base points have been chosen arbitrarily. So 5, does not 
leon, 
§ 5. We now pass to the case 9 = 3. 
We still denote by D,, D,,...., D, arbitrarily chosen points, 
1) Here by non degenerated is meant a curve not breaking up into a cz to be 
counted twice. In this manner is to be interpreted henceforth the expression non 
degenerated cg, used now and then. 
