From this ensues : 
In a pencil (c*) 
4 (n — 4) (n — 3)? 10n* + Bn! — 21n? — 80n + 20) 
curves have an injlectional point of which the tangent touches the 
curve in one other point more. 
§ 5. The locus of the inflectional points / of (ct) has a threefold 
point in each base-point and a node in each of the 3 (n — 1)’ nodes 
of the pencil, out of which we immediately find that tbe curve (/) 
is of order 6 (7 — 1) and of class 6 (n — 2) (4 n -— 3) *). 
Let us now deduce the order of the locus of the points V deter- 
mined by a e# on its inflectional tangents. 
As a base-point / lies on 3 (n — 3) (n + 1) inflectional tangents 
the curve (WV) passes with as many branches through B. So with 
an arbitrary c* it has 37° (2 — 3) (n + 1) + 3n (n — 2)(n— 3) points 
in common. 
Consequently (V) is a curve of order 3(7— 3) (n? + 2n—2). Now 
the curves (/) and (J”) have besides the base-points a number of 
points in common represented by 
18 (n — 1) (n — 3) (n° + An — 2) — In' (n — 3) (n + J). 
These points can only have risen from the coincidence of inflectional 
points with one of the points they have in common with the c” under 
consideration, thus from tangents with fourpointed contact. Such an 
undulation point, being equivalent to two inflectional points, is point 
of contact for (J) and (WV) from which ensues: 
5 ene) 
A pe a wm . » 5 
A pencil (c’) contains a (n 
3) (n° + n? — 8n +4) curves with an 
undulation point. 
§ 6. Let a threefold infinite linear system of curves c* be given. 
The ect osculating a right line / in the point P cuts the ray OP 
drawn through the arbitrary point O moreover in (n—1) points Q. 
The curves of (c"), passing through O form a net (er), determining 
on / the groups of an involution /,”. The latter having 3 (2 — 2) 
threefold elements, the locus (Q) passes 3 (n — 2)-times through O, 
so it is of order (4 — 7). 
Each of its points of intersection A with / is evidently a node on 
a curve of (ct), with / and OK for tangents. 
Each right line is nodal tangent for (An —7) curves of the system. 
From this ensues that the locus of the nodes K sending one of 
2) See Bosek, Casopis (Prague), XI, 283. 
49 
Proceedings Royal Acad, Amsterdam. Vol. VII. 
