1462 
tangent plane contains the associated asymptote of 4”; and two of 
those edges give therefore rise to 4 branches of the nodal eurve, 
which cut e‚ in Z, singly; the total number of these branches 
amounts therefore to: 
E (u— 2 — 20) (u—2e—26—1). 4. 
ty 
Through the edges of the second group coineiding in pairs pass 
2 sheets, which we can approximately realize if we suppose that 
two cylinders of revolution of which one lies inside the other rest 
on a table with the same edge. Let us suppose two pairs of such 
cylinders; each cylinder of one group cuts each of the other group 
along a curve with a node, because they have the same tangent 
plane; both the cylinders of one group and both of the other give 
rise to 4 curves of intersection, each with a node; i.e. the sheets 
of {2 passing through the edges of the second group, give rise, for 
each pair of these edges, to 8 branches of the nodal curve that 
each touch e, in Z,. The total number of these branches amounts 
therefore to 
4 o(o—1).8. 
Finally each sheet passing through an edge of the first group 
cuts the two sheets passing through an edge of the second according 
to 2 branches which both touch ¢, in Z, ; as, however, 2 sheets 
pass through an edge of the first group, each edge of the first group 
gives with each pair of coinéiding edges of the second rise to 4 
branches, which each touch e‚ in Z,; in total therefore: 
(u—2e—290). 6.4. 
If the three amounts found here are added up, we find that Z, 
is for the vest-nodal curve of $2 a (2u?—8ue—4pno—2u-+8e? + 
+8eo+4e+45*)-fold point. 
And from this it is in fact to be seen at once that the multi- 
plicity of Z, for the nodal curve is indicated by an even number, 
of which we have already made use higher up. 
For the general conic we tind from this 2.2° — 2.2 = 4, for the 
parabola 2.2? — 4.2 — 2.2 + 4 — 0, which agrees with the results of § 5. 
If the order of the rest-nodal curve is diminished with the multi- 
plicity of Z, we find the number of points that an arbitrary 
vertical plane outside Z, has moreover in common with that curve; 
these points are symmetrical in pairs with regard to 3, so that half 
of the number in question indicates the order of the projection of 
the rest-nodal curve out of 7, as centre on 8, and this projection 
is evidently the locus of the points that are centres of circles cutting 
