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while here all the branches have the asymptotes of k* as tangents (§ 7) ; 
3. boo simple points (§ 8); 4. 7 simple points lying in the points 
of contact of kv and |. And finally it possesses 4 x triple points, 
and on each cuspidal tangent of kh’ 2u-+-2r—de—2o—5 cusps whose 
tangents all coincide with the cuspidal tangent. 
§ 12. As, by the preceding investigations, the cyclographic surface 2 
has been completely inquired into, we must be able now to give 
an answer to any questions that may arise concerning the circles 
that cut A” perpendicularly. Let us therefore in the first place inquire 
after the curve that arises if we measure off on each tangent of 
ke from the point of contact a piece of prescribed length on either 
side; it is clear that we have simply to cut 2 with a plane that 
runs parallel with @, and that we have to project the intersection 
on 8. We find therefore a curve of order 2 (u—+ »— 2e — 6), which 
has nodes anywhere where it meets the rest nodal curve, and cusps 
where it meets the cuspidal edges of @. It further passes (»—o) 
times through the absolute points of 3, while it passes with 2 bran- 
ches, which each have the asymptote of 4” as an asymptote, through 
each of the «—2e—2o simple intersections of k* andl, and 
likewise passes with 2 branches through each of the 6 points of 
contact of #7 and /,, while those two branches touch here at 
ke as well. 
For the ellipse we find in that way a curve of order 8, consisting 
of two completely separated and closed ovals. The curve does not 
possess cusps, but does possess 8 nodes, 4 on the major axis and 4 
on the minor one. Of the 4 on the major axis 2 are real nodes, 
and they of course lie outside the ellipse, while the two others are 
isolated and lie between the foci; of the 4 on the minor axis 2 are 
likewise real nodes, and they of course lie again outside the ellipse, 
while the two others are imaginary in this case. Real points at 
infinity the curve does not possess at all, they appear with the hyper- 
bola where every time 2 branches also have as asymptotes the asy mp- 
totes of the hyperbola. For this, however, the nodes in one axis viz. 
the non-intersecting one, become all 4 imaginary, while in the cha- 
chacter of the 4 on the other axis no change arises; the 2 nodes 
lie now only between the vertices, and the two isolated nodes 
outside the foci. 
For the parabola the curve is of order 6; it possesses 2 real nodes, 
both lying on the axis of the parabola; one, a real node, lies 
outside the parabola, the other, an isolated node, between the focus 
and the point at infinity ; moreover 2 parabolical branches touch at 
Ll, in the point at infinity of the parabola. Further are, in the case 
