( 719 ) 
seroll of the right lines #3 the expression 1 (”—2)(n—4)(n? +-2n-4-12)+ 
+ n (n—A4) (82? +5n—24 , (a —4) (Tn—12). 
The principal tangents of @* whieh moreover touch the surface 
form a scroll of order n(n — 3) (n — 4) (nr? + 6n — 4). 
§ 6. The double tangents ¢ cutting ¢° in points D of the plane d 
form a scroll D, on which the section d” of 9" with dis a manyfold 
curve bearing } (7 n° + n+ 12)°) sheets. As moreover 
every double tangent of & belongs to (7 — 4) different points D the 
order of D is equal to 
3) (n—4) n° dn?) + bn en n—_3) (n-++3) (n—4) = 
n (n—1) (n+-2) (n—3) (n—4). 
According to § 5 n(n — 4) (n° + n° — dn — 6) double tangents c 
have one of their points of contact C in a given plane y and at the 
same time one of their points of contact D in the plane d. So this 
number indicates the order of the curve along which D and 9» 
touch each other. If we take the manyfold curve dr into consideration, 
it is evident that the points D’ which the right lines of D have in 
common with g” besides the points of contact C and the points of 
intersection lying in d, form a twisted curve (D’) the order of 
which in equal to 
nd Wet) (n—8) (n—4) — An (n—A4) (n?-+-n?—4n—6) — 
ie (n 4) n° dn) = $n (n—2) (n—A) (n—5) (An* H5n3). 
This curve evidently cuts d (n—4) (7—5)-times on each double 
tangent of dé”. In each of its remaining points of intersection with d 
the surface @” is touched by a right line, which is tangent to the 
surface in two more points. From this ensues : 
The points of contact C of the threefold tangents of o form a 
curve (C) of order $n (n—2) (n—A) (n—5S) (n?+5n+12). 
$ 7. On each right line c of the scroll D lie ( points D’ 
which can be arranged in 4 (n—5) (n—6) pairs D', D". If these pairs 
of points are projected out of an axis / by pairs of planes 2’, 4", these 
form asymmetric system, the characteristic number of which is $ 2(m—2) 
(u—4) (n—S) (2n?+5n-+3) (n—6). Each right line ¢ cutting / deter- 
mines a plane 4 evidently representing (n—5) (n— 6) coincidences 
a an 
A’ = 2". The remaining coincidences of the system (4) originate from 
coine eee. D = D", thus from threefold tangents d. As however 
1) In Cremona—Currze, Theorie der Oberfldchen, page 66 we find the expression 
} (n—3) (n—4) (n?+n—2) by mistake for the number of double tangents cutting 
g” in one of its points, 
