892 
for the nodal curve; this curve cannot have other points in common 
with |. So it cuts Ll in four tenfold points (i.e. the 40 points of 
before have changed into four tenfold ones) and so it is again of 
order 40 + 91 = 131. 
Also the surface @' undergoes considerable modifications as the 
conic lying in a plane 4 must now always touch the line /. The 
complex cone for a point P of / contains the ray /; the two tangen- 
tial planes through / to the cone coincide therefore; from which 
ensues that for each point P of 7 the two conics passing through it, 
coincide. The most intuitive representation of this fact is obtained 
by imagining instead of tbe point of contact of a 4’ with / two 
points of intersection lying at infinitesimal distance; if then on / 
we assume three of such like points, then through 1 and 2 passes 
a conic and through 2 and 3 an other differing but slightly from it, 
so that really through point 2 pass two conics. The loci of the 
conics is thus now again a 2* with nodal line 1, but this line has 
become a cuspidal edge, i.e. whereas formerly an arbitrary plane 
intersected 42! along a plane curve with a nodal point on /and only 
the planes through the four points S; (§ 15) furnished curves with 
cusps, now every arbitrary plane of intersection contains a curve 
with a cusp on / (and with a cuspidal tangent in the plane of the 
conic through that cusp). Furthermore we must notice that as the 
points 7;* coincide with S;, the four nodal points 7; will be found 
on the nodal line itself, thus forming in reality no more a tetra- 
hedron proper; nevertheless the property of the simultaneous cir- 
cumscription round about and in each other remains if one likes. 
18. The curve of intersection of order eighty of @* and 7° is 
again easy to indicate; it consists of the line / counted twelve times 
(for a cuspidal edge remains a nodal edge), and of a curve of contact 
of order 34 to be counted double ($ 15) which has with a plane 4 
through 4 fourteen points lying outside / in common and therefore 
twenty lying on /; these last however can be no others than the 
four points ‘;, for otherwise a generatrix of 2*° would have to 
touch a #° of 2* on J, which could only be possible (as / itself 
touches 4?) if a generatrix of 27° could coincide with / which is as 
we know not possible. The curve of contact of £&* and 2° passes 
thus five times through each of the four points S; which corresponds 
to the fact that five generatrices of 27° touch in S;, the degenerated 
conic (viz. the pair of points S;, 7;) lying in the plane /7;. 
The method indicated in § 14 to determine the number of torsal 
lines of the first kind undergoes no modification whatever; we can 
