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with the complete 1st polar surface by: 
2 (2’ = a’) (2? — Oro OU 
it appears at once by means of differentiation that z= 0 satisfies 
no more. ar, contains the curve /”, for the latter is a nodal curve 
for the complete first polar surface, as we saw in the preceding §. 
It further follows from the symmetry with regard to 3 that the 
tangent planes at a, are vertical in all the points of 4”, while this 
is also to be deduced from the fact that a straight line Z, P, con- 
necting Z with a point P of 4”, touches the surface a, in P, and 
so has here 8 points in common with the complete first polar 
surface 2, + B. 
_ This observation is important to us as we want to intersect ar, 
with the rest nodal curve of 2; all the points of the rest nodal 
curve namely that are lying in 3 and at the same time on 4” will 
as a matter of course belong to those intersections. But also the 
intersections of the rest nodal curve and > not lying on 4”, viz. the 
foci of Av, lie on z,. The first polar surface a, + 8 namely must 
contain the complete nodal curve, 2, therefore contains the rest 
nodal curve, consequently also the foci of 4’, and the vertical lines 
passing through these points have in those points three points in 
common with mr, + 3 and consequently two points with a, So we 
state that a, and the rest nodal curve have in common: 1. 2 (1 — 
— de —5)’ points, lying in the foci of k”; 2°) 2 (Sur + 3:—8e—3o) 
points lying in the vertices of k’. 
The rest nodal curve intersects >} further in the 2 (u —e— 2) 
(ry —2e—o) points P, which the tangents out of the two isotropical 
points have moreover in common with #? (ef. § 6); through each 
of these points pass 2 branches of the rest nodal curve, which both 
touch at the same vertical line, and 3 sheets of 2, which touch 
at the line ZP as well; from the latter we deduce that Z, P has 
in P in common with 26 points, conseqüently with a, + 3 5 points 
and with a, 4 points. If we now move the line 7, P a little, and 
do so parallel to itself, those 4 coinciding points diverge and arrange 
themselves into 2 pairs which are each other's image with regard 
to 8; from this it ensues that through P pass 2 sheets of a,, which 
both touch at ZP in P. Each branch of the rest nodal curve 
touches at those two sheets and consequently has with them together 
4 coinciding points in common; the two branches consequently 8 points, 
from which it ensues that the rest nodal curve and a, 3°) have in 
common 16 (u — & — 2) w — 2e — 0) points, lying in the intersections 
of the tangents out of the isotropical points at h* with ke. 
Let us consider a node JD of 4”. According to $6 there pass 
