(315) 
A, on which 4 is a fivefold line. Ten chords lying in every plane 
through J the scroll A is of order fifteen. 
Besides the fourfold curve FR; the scroll _/ contains a double 
curve of which we shall determine the order. 
If the points A; (¢=—1, 2, 3, 4, 5) lie in a plane with 7 then 
the fifteen points B =(4A:Ar, A/d) belong to the above mentioned 
curve. 
In order to find how many points B are lying on / we assign 
the point common to / and A; Az to the points common to / and 
the right lines A7An, Am4An and An4A,; hereby we create a corre- 
spondence (15,15) between the points of /. Two corresponding points 
only then coincide when a point B lies on /. In the correspondence 
there are still thirteen other points which differ from B agreeing 
with such a point; so B represents two coincidences. Hence / con- 
tains fifteen points & and the above mentioned double curve is of 
order thirty. 
3. If 4 has a point S in common with A; then dj; breaks up 
into the quartic cone, with centre S, standing on 2; and into a surface 
Aj, on which R; is a threefold curve, / remaining a fivefold line. 
Moreover by a very simple deduction it is shown that now the 
double curve is of order eight. 
4. If / becomes a bisecant 5 the surface 4); contains two quartic 
cones. The remaining scroll 4, has the fourfold line 5 and the 
double curve #;. The double curve (B) disappears here. 
By assigning each of the three points of A; lying with in the 
same plane to the chord connecting the other two, the chords of 
the scroll 4, are brought into projective relation with the points 
of Rs. 
So any plane section of df} is, just as &;, of genus unity and 
must have fourteen nodes or an equivalent set of singularities. This 
curve has five double points on B; and a fourfold point on 4. 
Evidently the missing three double points can only be represented 
by a threefold point derived from a threefold generator of 47, i.e. 
from the trisecant of the twisted curve. 
So a bisecant will be cut only by one trisecant. 
5. As 5 meets in each of its points of intersection with the 
curve two trisecants, the trisecants of Ls; forma scroll T; of order five 
of which ZR; is a double curve. Evidently Z; can have no other 
double curve, so this surface is also of genus unity. 
