105% 
(n—A4) points S arising from the er having a as tangent ¢,. In each 
of the remaining intersections a point R, has coincided with a point 
S into a point R;. The points, where a c* possesses a five-point tangent, 
lie therefore on a curve (R‚) es order (40n—105). 
For n= 4, the number 55 is duly found. 
3. To each cr possessing a tangent ¢,, we associate that tangent ; 
the latter cuts it moreover in (n—5) points |. The locus of the 
points |’, together with the curve (/?,) to be counted five times, is 
produced by the He eae systems [c"] and [¢,]. The system [¢,| 
has (§ 1) as index 10n(n —4). The curves c” passing through a point 
P form a net; in this net occur 15(-—4)(4n—5) curves with a ¢, '); 
this number is the index of [c"|. For the order of the curve (V) is 
found 15 (n — 4) (dn —- 5) + 10n 2(n — 4) — 5(40n — 105) = 5(n — 35) 
(21° + l4n — 33). 
In the pencil of rays (JM) the pairs of points &,,V determine 
a correspondence with characteristic numbers (n—5) (40n—105) and 
(n—5) (10n?+70n—165). The 10n(n—4) tangents ¢, passing through 
M produce each (n—5) coincidences. As the remaining ones arise 
from the coinciding of R, with V, it appears that I contains 
3O(n—5) (5n—9) curves with a sia-point tangent ty. 
The symmetrical correspondence (MV, MV") has as characteristic 
number (102?+70n—165) (n—5) (n—6), while the 10n(72—4) tangents 
t, passing through Jf represent each (n—5)(n—6) coincidences. 
From this ensues that I possesses 10 (n—5) (n—6) (n?+18n-— 33) 
curves with a tangent t5,2. 
4. The Z?, which I determines on /, possesses 6(n—3) (n—4) 
groups in which a triple element occurs beside a twofold one; 
consequently is / for 6 (n—98)(n—4) curves a tangent f23. If / rotates 
round P, the points of contact AR, and R, describe two curves 
(Re)o3 and (Ra)o3. P is as base-point of a net, point of contact Ff, 
for 3 (n— 4) (n+3)’), point of contact Rk, for (n—4) (n-+-9)*) curves. 
So (Ro)o2 is of order (n—4)(8n-++-9)+-6(n—3)(n—4) or (n—4)(9n—-9) 
and (Rs)23 of order (n—4) (n-+-9)-+-6 (n—3) (n—4) or (n—A) (7n—9). 
From the correspondence (MR, Wf,) may be deduced again that 
t, envelops a curve of class 10n (n—4). (See $ 1). 
Each tangent fez passing through P cuts the corresponding c* in 
(n—5) points W; on a ray passing through P lie therefore 6 (n—3) 
70 
Proceedings Royal Acad. Amsterdam. Vol. XVII. 
