(785) 
$ 4. Returning to the scroll B ($ 2) I consider the points of 
intersection of the twisted curve (5') with the plane @. Each point 
of intersection of ¢” with an inflectional tangent lying in @ can, be 
regarded as the point B, each one of the remaining (7 —4) as a 
point 5'. Hence the curve (B) meets 3u (n — 2) (n — 3) (n — 4)-times 
Br on the inflectional tangents of 3. In each of the remaming points 
of intersection of (B) with 3 we find that &* is touched by a right 
line having elsewhere three coinciding points in common with 9”. Such 
a right line is called by me a tangent 423, A being its point of 
osculation, B its point of contact. 
The points of contact of the tangents tas form a curve of order 
n(n — 2) (n — 4) (n° + 2n + 12). 
$ 5. In each point C of the curve y” according to which 9” is 
cut by the plane y I shall regard the (n— 3) (n + 2) tangents ¢ which 
touch pr moreover in a point €’. On the seroll C of the double 
3) (n+2) 
sheets meet. Each double tangent situated in y representing two right 
tangents c the curve y” is a manyfold curve in which (7 
lines of C the order of this seroll is equal to 
n(n—)(n + 2) + n(n — 2) (n — 3) (21+ 3) or n(n —3) (n° + An — 4). 
The surfaces g and C touch each other along the locus (C’) of 
the two points of contact. Of this curve the plane y contains the 
points of contact of the right lines ec lying in y besides the points 
C=C", where a right line ¢ is a fourpointed tangent. So the order 
of (C’) is n (n—2) (n?—9) + n (lln—24) or n(n? — 2n? + 2n — 6). 
Besides the curve (€’) to be counted twice and the curve y” to 
be counted 2(n— 3) (n + 2)-times C and ¢ have moreover in 
common the locus of the points S determined by the double tangents 
c on $”. The curve (S) is of order n° (n — 3) (n° + An — 4) — 
2n(n?—2n? + 2n—6) — 2n(n—3) (n-+2) or n(n—4) (n° +- n?—4n—6). 
To the points of (S) lying in y belong the points of intersection 
of y" with its double tangents c. As each of the two points of contact 
of ¢ can be regarded as point C these points of intersection ‚S must 
be counted twice. The remaining 2 (7 — 4) (n° + n° — An — 6) — 
n(n —- 2) (n? — 9) zw — 4) points S lying in y are apparently points 
of osculation of the tangents 423. So from this ensues: 
The points of osculation of the principal tangents touching @* 
moreover elsewhere form a curve of order n (n — 4) (38n? + 5n— 24). 
The curves (A) and (4) formed by the points of osculation and 
the points of contact of the tangents 43 have the points of contact of 
the fivepointed tangents in common. Taking this into account we 
find (by again projecting out of an axis /) for the order of the 
