Mathematics. — Double points of a c, of genus O or 1. By 
Dr. W. van per Woupe. (Communicated by Prof. ScHourr.) 
(Communicated in the meeting of November 26, 1910). 
ae | 
§ 1. A curve of order 7» is in general determined by > n(n + 3) 
single conditions. So a curve of order five is determined by 20 points, 
or — a node counting for 3 single data — by 6 double -points and 2 
single points. However, it is not possible to take arbitrarily the double 
points of a rational curve of higher order than the fifth; for the 
1 
number of double points of this curve amounts to 5 (n—A) (n—2) 
1 
and for n> 5 three times this amount is larger than — n (n + 3). 
ad 
Between the double points of a curve of order 2 whose number of 
1 
double points is greater than ee (2 + 3) one or more relations must 
exist. “So far no attempt seems to have been made to express those 
relations geometrically” ;*) in the following paragraphs I intend to 
do this with reference to the curve of order six. 
§ 2. A curve of order six is determined by 27 single data, hence 
by 27 points. It can possess at most 10 double points; according to 
the preceding 9 of these can be taken arbitrarily and really a curve 
of order six is determined by 9 double points taken arbitrarily; this 
is however degenerated into a cubic curve to be counted double 
through those 9 points. 
If therefore a (non-degenerated) curve of order six has 9 double 
points, there must be a relation between these already ; only 8 can 
be taken arbitrarily. In future we shall understand by D,, D,,.,.D, 
points taken arbitrarily; the locus of the point forming together with 
these a system of 9 double points of a curve of order six not degene- 
rating into a cubic curve to be counted double we shall for the 
present represent by 7, whilst by c, an arbitrary curve of order 
n is meant. 
§ 3. If D, is a point of j, there exists a (non-degenerated) curve 
of order six possessing a double point in each of the points D,, D,,... D,; 
furthermore we can lay through these 9 points a cubic curve. If these 
1) SALMON FreprER, Hohere ebene Kurven, p. 42. 
41 
Proceedings Royal Acad. Amsterdam. Vol. XIII. 
