1066 
involution formed by the pairs d,d’. So K is node of the locus (C) 
of the cusps. 
All er passing through an arbitrary point P form a net, N. The 
curve J of N has a node in B and passes through all the points 
K; for through B passes one c* of the pencil of nodal curves deter- 
mined by A. The curves (C) and J have two points in common 
in each point A; they further intersect in the 12 (n—1) (n— 2) cusps 
of N; the remaining intersections are found in B. From 4 (2r—3) 
(8n—8) —- 2 [6(n—1)* —4] —12 (n—1) (n—2) = 8 it appears that the 
curve (C) has a quadruple point in B. 
B is node for all c* of a pencil, consequently cusp of two c*; 
from this follows that each of the two cusptangents is touched by 
two branches of (C). 
Any point K is fleenode for jive cv. In order to understand this 
we consider the curve which arises if to each nodal c” of the pencil 
(K) its tangents d,d’ are associated. The crt? thus produced has 
with a line d only (n—8) points outside A in common. 
The locus (EF) of the jlecnodes passes therefore five times through 
each of the critical points. 
The locus J of the nodes in the net MN, which is set apart from 
r by an arbitrary point P, has with (/’) five intersections in each 
point K. They further have the 3 (m—1)(10n—23) fleenodes *) of 
N in common; the remaining intersections lie in the 5 base-points. 
From 3 (n—1 ) (20n—383)—5| 6(n—1)*—46 |—-3 (n—1) (10n—23) = 206 
it appears that the curve I passes ten times through each of the 
base-points. 
Each of the inflectional tangents f of the five c”, having a flecnode 
in B, touches two of the branches. 
18. The curves (C) and (/’) have in the critical points AK and 
the base-points B of FP 10[6(n—1) — 46] + 406, or 60 (n—1)’ 
points in common. Each of the remaining (202 —383) (82—12)—60 
(n—1)* intersections is a cusp with a four-point tangent and at the 
same time to be counted twice as fleenode. /n V occur therefore 
(50n?—192n-+168) cusps with four-point tangent. 
If we have n=3, b==6, these particular curves are easy to 
determine. Any line BB’ is tangent of two conies passing through 
the remaining four base-points; through each point 6 pass two 
tangents to the conic of the remaining five base-points. All in all 
IN p. 944. 
