500 
5. As we have seen before 4! is for the surface 2° a single curve, 
k® a nodal curve, and the surface cannot contain other nodal curves 
for, if a point U is to be a double point, then through this point either 
more than one chord a or more than two chords 6 must pass; the 
former is only possible for the points of 4°, the latter only for those 
of k*, and these two curves we have already investigated. On the 
other hand the surface contains a number of single lines crossing 
each ether, as many as twenty; the chords of #* namely form a 
congruence of rays (1,3), those of £* one (2,6), and these congruences 
have according to the theorem of HaArpueN 1.2 + 3.6 = 20 rays in 
common. Through a point P of such a ray passes one chord a, one 
chord 6 coinciding with a and one chord 46 more; so it is a single 
point for 2°. Two of these lines cannot possibly intersect each other 
outside %*, for in that case two chords « would pass through one 
point, which is impossible; it is not impossible for them to intersect 
on #°, but this requires a peculiar situation of 4° and £* with respect 
to each other, which we will not presuppose. 
An arbitrary plane through one of the twenty lines cuts 2° 
besides in this line still according to a curve of order five which 
has with the line in common its two points of intersection with 4% 
but not those with 4, because the latter are but single points for 
the surface. However besides the two points of intersection on 4° 
the curve must have three points more in common with the line, 
in which points the indicated plane must therefore touch the surface; 
so the surface 2° possesses an infinite number of threefold tan- 
gential planes, which are arranged in twenty pencils of planes, around 
the twenty lines of the surface as axes. 
7.8.9 
— — 1 = 83 points or 
1.2.3 
in general single conditions; we shall investigate for how many 
single conditions 4°, 4*, and the twenty lines of the surface count. 
The curve /* must, be a nodal curve; so we try to construct a surface 
of order 6 having &* as a nodal curve. In an arbitrary plane a we 
assume eighteen points quite arbitrarily ; we determine the three 
points of intersection of « with £°, and we construct a plane curve 
of order 6 having these last three points as double points and at 
the same time containing the 18 points above mentioned; as a double 
point counts for three single data and a curve of order 6 is deter- 
mined by 4.6.9 = 27 points, we have in a just enough data to 
determine the curve of order six. 
In a second plane 3 we assume arbitrarily only 12 points, 
and we add to these the six points of intersection with the curve 
A surface of order 6 is determined by 
