940 
whilst w,, (3), B, and 7 keep the signification assigned to them in 
art. 4. Now the question is to determine the locus of the point 
forming with D,, D,,....., Dp a set of threefold: points-of a non 
degenerated ¢,. 
In order to determine a curve c, passing three times through 
D,, D,,...., D, we can imply to it the condition of containing six 
arbitrary points. Of these six points however no more than two ’) 
may lie on w,; then the last point common to w, and c, is deter- 
mined inequivocally. We will show immediately how the latter point 
can be found; provisionally we start from any c, with the eight 
given threefold points, cutting c, in an arbitrarily chosen fixed point 
XY. This c, cuts uw, in two points more; the line connecting these 
two points has still a third point / with , in common; according 
to the Residual Theorem of Syivester the latter point isa fixed point, 
i.e. independent from the chosen curve e, passing through X. Now 
we first determine the point £; to that end we choose ac, breaking 
up into a curve v, of pencil (8) and a curve c, passing twice 
through D,, D,,...., D, and passing moreover through X. We have 
seen that this c, cuts uw, in one point } more, being collinear 
with X and 7 ($ 4, III); moreover uw, and v, have B, in com- 
mon. So the point # is the third point of intersection of the line 
FB; “and 2s, 
If now we fix on u, two points X, X’ and consider a curve c, 
with threefold points in D),, D,,...., D, and cutting uw, in X and 
X’, then the last point of intersection of this c, and w, can be found 
as follows: we first determine in the manner indicated the point Z; 
then the third peint of intersection of the line HX’ and u, is the 
point looked out for. 
temark. We have stated, that any c, with D,, D;...., Di as 
threefold) points meets w, in three points more; evidently this does 
9 
not hold if this ec, breaks up into two curves one of which coincides 
with w,. In this case the residual curve of order six must be deter- 
mined in such a manner that it admits on w, nine nodes, eight of 
which lie in D,, D,,....,D,;: So we fall back on the case 9 = 2: 
but we can diseard this by requiring that D, has been determined 
in such a way tbat the c, under discussion does not break up, neither 
into a c, to be counted thrice nor in two curves c, and ¢,, the latter 
of which admits a node in any of its points of intersection with 
the former. 
Ur 
ge 
SALMON FtepLER: Höhere ebene Kurven, p. 25. 
