( 630 ) 
two curves are represented respectively by the equations , = 0 
and u‚==0, then by w‚ +4wu?,=0 a pencil of curves. of order six 
is determined, each of which possesses in each of the base points of 
the pencil a double point. Hence 
If D,, D,...D, are double points of a curve of order sia not 
degenerating into a cubic curve to be counted double, these points are 
the base points of a pencil of curves of order six with nine common 
double points. 
For shortness’sake we shall in future call such a pencil a c, 
pencil with 9 double points; of course through each point passes one 
curve of this pencil. 
§ 4. Out of the c, pencil determined by the base points D,, D,...D, 
we choose one w,; we furthermore introduce a variable c,, possessing 
double points in D,, D,...D,, but being for the rest undetermined. 
These two curves intersect each other in 2 more points; the line 
connecting these two points intersects uw, in another point 7. This 
last point is according to the Residual Theory of SyLvestmr a fixed 
point, i.e. independent of the c, chosen by us. Point 7’ is easy to 
determine; it is evident that if we take for c, a c, counting double, 
T is the tangential point of B,, the ninth base point of the c, pencil 
rouke ts 
If of a pencil c, the double points D,, D,...D, and a point P 
are determined, then according to the preceding another point P’ of 
c, is determined; if namely we lay through D,, D,...D, and Pa 
cubic uw, and if we determine on it the tangential point 7’ of B,, 
then the third point of intersection of FP with u, is the point P’ 
under discussion. Farthermore it is evident from this that only these 
points of wu, having 7 as tangential point can be the ninth double 
points of a c,, which has already double points in D,, D,...D,. If 
however we impose the condition that this c‚ may not have degenerated 
into a c, counting double, then one of these points, viz. B, does not 
count; for, there is not.a single non-degenerated c, which has double 
points in D,, D,...D, and B,. For then we should be able to 
bring a c, through these points and a point chosen arbitrarily on 
the curve c, and the curves c, and c, would have nineteen points 
in common. If however J, /’, and J" are the other points of u, 
having 7’ as tangential point, then each c,, having double points in 
D,, D,... D, and passing through one of these points, will have 
there two points in common with u,. 
$5. We can determine ac, pencil by the double points Di). RDE 
