( 713 ) 
O one of the points S lies in O. So the curve (S) ús of order 
(n +1) (n — 2) + 2 (n — 1) (n — 2) = (rn — 2) Bn — DV. 
If B is a base-point of (cn), then only 2 (n — 2) points 7’ (the double 
points of an /*—!) lie on OB outside O and B. So OB touches in 
B the tangential curve of O whilst it is (m — 2)-fold tangent of ($). 
Each of the 2 (n — 2) eurves c* touching OB projects a point Sin B. 
So each base-point is a 2 (2 — 2)-fold point of the satellite curve. 
The common points of the tangential curve @"—! and the satellite 
curve sr?) form four groups. 
First there are (n + 1) (n — 2) united in O. 
Secondly 2 (n — 2) lie in each base-point B. 
Thirdly the two curves touch each other at each inflectional point 
sending its tangent through 0. 
‘Fourthly they cut each other in the points of contact of each 
double tangent passing through O. 
Now the inflectional tangents of a pencil envelop a curve of class 
dn (n — 2). *) 
So the number of points of contact of inflectional tangents through 
O amounts to 
(n — 2) (Bn — 1) (2n — 1) — (n — 2) (n + 1) — 2 (n—2) n? —6n (n—2) = 
= 4n (n — 2) (n — 3). 
The double tangents of the curves c* belonging to a pencil envelop 
a curve of class 2n(n — 2) (n — 3). 
§ 3. Following Emi. Weyr’) we consider the curve ct! gene- 
rated by the pencil (c") with the pencil projectively conjugate to it 
of the tangents in a base-point B. As each c” cuts its tangent more- 
over in (2 — 2) points, B is a threefold point of the c’+!. From this 
ensues easily that through B can be drawn (n + 4) (n — 3) tangents 
to etl, As many double tangents of the pencil (ct) have one of 
their points of contact in B. 
We shall now consider the satellite curve of B. On each ray r 
through B lie 2(m— 2) points of contact 7, so 2 (n — 2) (n — 8) 
points S. If # coincides with one of the double tangents just mentioned, 
one of the points S lies in B. So B is an (n+ 4) (n — 3)-fold point 
on (S) and the order of (S) proves to be equal to (n + 4) (n — 3) + 
2 (n -— 2) (n — 3) = 3n (n — 8). 
The tangential curve of B has in B a threefold point; for a ray 
1) For this is the number of tangents of pe 
of OB can be drawn through O. 
°) Sitzungsberichte der Akademie in Wien LXI, 82. 
which besides the 7? tangents 
