726 
£6 2 AOS 112. 
This number comprises all pairs of lines of the surface whose 
components lie at infinitesimal distance and intersect each other, 
thus the 6 double generatrices, the 48 torsal lines of the first kind 
and the still unknown number of torsal lines of the second kind: 
so 2° contains 58 torsal lines of the second kind. 
For a congruence is characteristic, besides the number of rays 
through a point (in our case 6) and in a plane (in our ease 15), 
the number of pairs of rays which belong with an arbitrary line 
to a pencil, the so-called rank; according to the preceding this number 
is nothing but our quantity «, thus 40; the congruence deduced from 
Q*° js therefore a (6, 14, 40). 
The results found above allow being controlled, by our finding 
the 4131 = 524 points of intersection of the surface 2* with 
the nodal curve of 2°. The greatest number of these points we find 
united in the points 7; and 7;*, the eight nodal points of 2*. A 
point 7; is a 36-fold point of the nodal curve (§ 13) and counts 
thus for 72 points of intersection; a point 7;* is a tenfold point of 
the curve and counts thus for 20 points of intersection, together 
4 >< 92 = 368. In the 40 points where the nodal curve rests on /, 
the curve meets the double line of 22%; so this gives 80 points. Ina 
pinch point of a torsal line of the second kind the nodal curve traverses 
2* in a single point of intersection. Let us assume e.g. a plane 2 
through 7 and such a torsal line as well as two planes 4, and 2, on 
both sides of 2 and in the immediate vicinity of 4; then in 4, eg. 
two generatices of 2°° will nearly coincide, so their point of inter- 
section will almost lie on the conie of @* lying in this plane; in 4 
itself this point of intersection really falls exactly on #°, and in 2, 
the two tangents have become conjugate imaginary; their point of 
intersection has nevertheless remained real, i. e. the nodal curve 
naturally continues its course but now lies inside 4’; so it has inter- 
sected the surface. As @°* possesses 58 torsal lines of the second kind 
we find 58 new points of intersection. 
We must finally discuss the 6 double generatrices of 2*° which 
bear themselves as regards the nodal curve about the same as torsal 
lines of the second kind do. We must not lose sight of the fact that 
a double edge d of 2° is a singular ray for the congruence but not 
for the complex; so if it intersects / in D, then the complex cone 
of D shows in no way anything particular; the plane 2 through 
l and d contains thus two different generatrices of that cone, of 
which d is one. The consequence is that the conic of @* in 2 must 
touch the line d in some point or other not lying on /, through 
