119 
apart for reasons more than once mentioned, then there remain 
twenty; the rays s corresponding to these rest on / as well as on om, 
i.e. the rays resting on / form a surface 27°. This in order to 
control the result. 
§ 14. We shall now try to determine the order of the nodal curve 
of 2°, which is according to the preceding equal to 91, augmented 
by the number of points unknown for the present, with which that 
curve rests on /; this number is connected with other numbers 
which we must also calculate to be able to find the former, and to 
this a deeper study is necessary of 2*°, as well as of the figures 
which are in relation with this surface. 
A scroll possesses in general a certain number of pinch points and 
torsal lines, and those of @°° can be divided into two kinds which 
bear themselves very differently in the following considerations. To 
the first kind we reckon the torsal lines whose pinchpoint lies on / 
but whose torsal plane does not pass through /; to the second kind 
the dualistically opposite, thus those whose pinchpoint does not lie 
on 7 (thus on the nodal curve to be investigated), but whose torsal 
plane for it does pass through /. 
A third kind might be a combination of the two others, torsal- 
lines, whose pinchpoint lies on / and whose torsal plane passes 
through /; we shall however show that these do not appear on 22°. 
We can get some insight in the appearance of these torsal lines if 
we return to the regulus and the curve £* of the preceding §; 
k** contains the foci of all generatrices s of °°, and the regulus is 
the locus of all the rays s, which are conjugated to the points P 
of /. Moreover lie in a plane throngh / fourteen generatrices of 22° 
and the foci of these lie on a cubic curve through the four cone 
vertices. Let us now consider the generatrices of the regulus and the 
curve &**. A generatrix s, of the regulus intersects @° in six points 
and these lie on £'°, for 4°? is the intersection of 2° with theregu- 
lus; the rays s corresponding to these six points are the generatrices 
of 2°, which pass through a same point P of /, viz. the focus of s,. 
If however s has two coinciding points in common with #'°, then 
two of the six generatrices through P coincide, and this can 
Lappen in two ways. The curve £'* has namely 6 nodal points (viz. 
on 4%, and through each of these passes a line s, which has with 
k'* besides the nodal point only four points in common; of the six 
generatrices of £2°° through the focus P of s, two coincide and that 
in a double generatrix of 42°, the number of which, as we know, 
($ 13) amounts to 6. Those double lines can be regarded as “full 
