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through this point 4 branches of the rest nodal curve, and 4 sheets 
of 2, which all touch at the line ZD in D; Z, D has therefore 
in D with 2 8 points in common, consequently 6 points with z,, 
and by applying again, as above, the proceeding of the parallel 
shifting, we find that through D 3 sheets of a, pass, which all touch 
at the line ZD. Each of the 4 branches of the rest nodal curve 
which pass throngh JD, touches at each of these 3 sheets; this 
procures in total 24 coinciding points, so that we can say: the 
rest nodal curve and a, have 4°) 24d points in common, lying in 
the nodes of ke, 
Let us consider a cusp A of 4’. Through an arbitrary point of 
kv pass 2 tangents less than through a point that does not lie on 
ke, and as with each tangent 2 generatrices of £2 correspond, the 
vertical passing througb an arbitrary point of 4 has 4 coinciding 
points in common with @, and passing through a node & (vide 
supra). Through a cusp pass three tangents less than throngh an 
arbitrary point of the plane; therefore the vertical 7, A has in A6 
points in common with @, and consequently 4 points with zr, 
Through K pass 2 cuspidal edges of 2, and we know (ef. $9) that 
they are nodal edges for 2, and consequently simple edges for =,,. 
Let us therefore imagine a vertical in the neighbourhood of Z, K, 
cutting the two cuspidal edges, the latter has then on those cuspidal 
edges already 2 points in common with 7,, and consequently quite 
close to them 2 more other points, which are each other’s image 
with regard to 3. From this we infer that through A’ too, pass 3 
sheets of z,, viz. 2 through the two cuspidal edges and still a third 
formed by those two other points and touching at the line 7, K, 
because it must have 4 points in common with a,; and as through 
K pass 6 branches of the rest nodal curve and that without vertical 
tangents, there lie in A 18 intersections of the rest nodal curve and 
ax, united. Consequently: the rest nodal curve and z, have 5°) 18 « 
points in common, lying in the cusps of k*. 
With this the intersections of the rest nodal curve and z,, as far 
as they lie in 3, have been summed up. 
The rest nodal curve and 2, have also points at infinity in common. 
In the first place Z, has as a point of a, the same character as 
as a point of 7,, and as a point of @, so that all the points of the 
rest nodal curve, of which we calculated in § 5 that they lay in 
Z, Or infinitely near to it, also belong to z,. This number amounted 
to ($ 8) 2u? — 8ue — Qu + 8s? + de — 4s, and this would therefore 
be the number that we have in view here, if zr,, as ¢,, had a simple 
point in Z,. This, however, is by no means the case, as we observed 
