935 
Mathematics. — ‘Characteristic numbers for nets of algebraic 
curves.” By Professor JAN pr VRIES. 
(Communicated in the mecting of November 28, 1914). 
1. The curves of order n, c”, which belong to a net N, cut a 
straight line / in the groups of an involution of the second rank, 
I,,°. The latter has 3 (n—2) groups each with a triple element); / 
is therefore stationary tangent for 3 (n—2) curves of XV. 
Any point P is base-point of a pencil belonging to .V, hence 
inflectional point for three curves’) of NV. 
The locus of the inflectional points of MN which send their tangent 
i through P, is therefore a curve (Ip of order 3 (n—1) with triple 
point P. 
If the net has a base-point in B any straight line through 5 is 
stationary tangent with point of inflection in B. Consequently (D)p 
passes through all the base-points of the net. 
We shall suppose that MN has only single base-points. 
On PB WN determines an /?_,; the latter has -3 (n—3) triple 
elements; from which it ensues that B is an inflectional point of 
(/)p having PB as tangent 7. 
Through P pass 3 (n—1) (2n—38) straight lines, each of which 
touch a singular curve in its node‘); all these nodes D lie apparently 
on (/)p. 
2. Every c”, which osculates / in a point /, cuts it moreover in 
(n—3) points S. We consider the locus of the points S, which belong 
in this way to (/)p. Since P, as base-point of a pencil, lies on 
3(n—3) (n+-1) tangents of inflexion‘), the curve (S) has in Pa 
3 (n—8) (n+1)-fold point. Apart from P each ray of the pencil (/) 
contains 3 (n—2)(n—3) points S; hence (S) is a curve of order 
3 (n—3d) (2n—1). 
Let us now consider the correspondence between the rays s and 
s, which connect a point Jf with two points S and Z belonging to 
1) If the J,? is transported to a rational curve c* and determined by the field 
of rays, these groups lie on the stationary tangents. 
2) For the characteristic numbers of a pencil my paper “Faisceaux de courbes 
planes” may be referred to (Archives Teyler, sér. II, t. XI, 99—113), For the 
sake of brevity it will be quoted by 7. 
8) Cf. for instance my paper “On nets of algebraic plane curves”. (Proceedings 
volume VII, p. 631). 
4) T. p. 100. 
