1067 
we find therefore 15x 2+6x2= 42 figures (c’, c’), satisfying 
the condition. 
The tangents in a fleenode we shall indicate by f and d; f indi- 
cates the inflectional tangent. We shall determine the index of the 
system [d |. 
The curve (D)p, containing the nodes which send one of their 
tangents through YP (§ 12), passes through the points A’ and twice 
through the points 6. With (/’) it has, apart from those points, 
(4n—-5)(20n— 33)—30(n—1)?_ or 50n?—172n+135 intersections. As 
many tangents f and d pass through ?. The number of lines / 
amounts according to $15 to (10n’—32n+-18), hence [d | has as 
index (40?—140n--117). f 
In order to find the locus of the points G, which any flecnodal 
en has in common with its tangent d we consider the product of 
the projective systems [c” | and [d |. Their indices are 3(n—1)(10n—23), 
i.e. the number of fleenodal c” in a net, and (40n? — 140n + 117). 
Since the curve (/’) belongs three times to the figure produced, we 
find for the order of (G) 3(¢n—1) (10n —23) + n(40n?—140n+-117)— 
—3(20n—33) or (40n°?—110n?—42n+168). 
Let us now consider the correspondence (MF, MG). The straight 
lines d passing through MZ produce each (n—3) coincidences; the 
number of the remaining ones amounts to (202—33)(n—3)+(40n?— 
110n’—42n+-168)—(40n?—140n-+-117)(2 —3)—=170n? —672n+ 618. 
To the coincidences “= G determined by this belong in the first 
place the cusps with tangent ¢,; the remaining ones arise in pairs 
from nodal curves with two inflectional tangents f. Their number 
amounts therefore to 4 [(170n°—672n4618) — (50n?—192n+-168)] ; 
the complex contains (6On? — 240n + 225) curves with a flejlecnode. 
In the case n=3, b=6 we find 45 for it. Each of the trilaterals 
belonging to I’ is apparently to be considered as a figure with three 
flefleenodes. 
19. In a similar way as in N$5, 15, 14 it may be determined 
how many times a base-point 4 of the complex is point of contact 
of a particular tangent. We find then in the first place that B is 
point of contact A, of ten tangents ¢,. It is further consecutively 
found that B is point of contact ft, of (n—5) (n-+-16) tangents 44,9, 
point of contact 2, of 3 (n—5) (n+6) tangen:s ¢s3 and of 2 —5) 
(n—6) (n+6) tangents 4322, point of contact A, of 2(m—5) (8-8) 
tangents fas, of 3(n—5) (n—6) (3n+8) tangents fo39, and } (n—5) 
(n—6)(n—7)\(3n-+-8) tangents f2209. 
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