937 
‘well-known theorem, according to which a net of eubies contains 
21 figures, composed of a conic and a straight line. 
5. To the intersections of a with the curve (5), belong the 38(—2) 
groups of (n — 3) points S, arising from the curves c”, which 
osculate a. In each of the remaining 3(n?-+-n—5) — 3(n—2)(n—3) 
intersections a point / coincides with a point S of one of the three 
cr, which have / as point of inflection. The corresponding tangent 
{then has in common with c* four points coinciding in J, so that 
I is point of undulation. Zhe points of undulation of the net he 
therefore on a curve (U) of order 3 (6n — 11). 
For n= 3 we find the 21 straight lines belonging to the degenerate 
eubies of the net. 
As a base-point B of a net is point of inflection of oo! curves 
cr, there will have to be a finite number of curves, for which 5 
is point of undulation. In order to find this number we consider 
the locus of the points Z’ which any ray ¢ passing through B has 
still in common with the c", which osculates it in 5. As Bis point 
of inflection on three c* of the pencil which has an arbitrary point 
P as base-point, the curves of N falling under consideration here 
form a system [c”| with index three, which is projective with the 
pencil of rays (¢). 
The two systems produce a curve of order (2 + 3), which is cut 
by a ray ¢ in (n— 3) points 7. Consequently it has in 6a sextuple 
point, and there are six curves c”, on which JZ is point of undulation. 
If the net has base-points they are sixfold points on the curve (U). 
For n= 3 the curve degenerates into a sixray, which consists of 
parts of compound curves. 
6. To each c”, which possesses a point of undulation U we shall 
associate its fourpoint tangent w; the latter cuts it moreover in 
(n—4) points V. The locus of the points forms with the curve (U) 
counted four times the product of the projective systems |c"| and 
(u). In the pencil which a point P sets apart from MN occur 
6(n—-3)(8n—2) curves, which possess a point U*); this number is 
therefore the index of {c"|. The system [u] has, as appears from 
§2, the index 6n(n—3). In a similar way as above ($ 4) we find 
now for the order of (V) 6(n—3)(8n—2)+6n?(n—-3) —12(6n—11) = 
=6(n— 4)(n? +4n—7). 
We now associate on each straight line w the point U to each 
1) T. p. 105. 
62 
Proceedings Royal Acad. Amsterdam. Vol. XVII. 
