1165 
For the number of these points we find therefore y (y + 1)? — 
ANG deken met PB 
Now the surface (R)/+' has in common with (/) besides the 27? 
points lying on o and the null-points of the plane PQR still y (Q—y) 
more points and this is only possible for either y=1 or y= 2. 
So we may conclude that for y > 2 the singular points must 
be arranged at least on two curves. 
Mathematics. — “On plane linear null-systems”. By Prof. JAN Dr 
VRIES. 
(Communicated in the meeting of January 25, 1913). 
$ 1. By a plane null-system (a, 8) we understand a correlation 
between the points and lines of the plane in which to any point 
F correspond «a null-rays f passing through it and to any ray f 
correspond @ null-points situated on it. 
We restrict ourselves to the case a= 1 in which any point F 
bears only one null-ray (near null-system) and represent by A the 
second characteristic number. 
If the ray f rotates around a point P, its & null-points describe a 
curve of order 4-+1 passing through P and touching in P the 
null-ray of P; we denote that curve by (PH. 
The curves (PF! and (Q)*+! have the & null-points of PQ in 
common; any of the remaining (& + 1)? — 4 points of intersection 
bears a ray through P and another ray through Q, therefore a 
pencil of null-rays; so these points are singular. 
Therefore a null-system (1, k) admits 4? 4 k 41 singular points. 
The curves Pt! form together a net with 47 -+ + 1 base points; 
through any pair of arbitrarily chosen points X, Y passes one curve 
determined by the point common to the two null-rays 2, y. 
A pencil of curves yg” with n° base points determines a linear 
null-system, in which to any point / corresponds the tangent J in 
f to the curve passing through #. This pencil intersects an arbi- 
trary line f in the groups of an involution of order n, admitting 
2(n—1} double points, therefore 4—=2(n—1). This null-system 
admits (4n’—6n-+ 3) singular points. To these belong the n? base 
points, lying on o' tangents; the remaining ones must be nodes 
of curves g”. So we fall back on the known property of the pencil 
(p) to contain 3(z—1)’ curves possessing a node. 
