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of the parabola, the circle points, simple points, in the case of the 
two other conies, nodes. 
If the plane of intersection is placed in an oblique position so that 
it gets an intersection ¢ in common with 8, the circles are found 
that cut 2” perpendicularly and d under an angle of constant cosine, 
which cosine may very: well be > 1 (viz. if the angle of the plane 
with 8 is < 45°); if the angle is exactly 45° the circles cutting 4? 
perpendicularly and touching at d are found. 
The circles cutting 4” perpendicularly and passing through a given 
point P of @ (which point may or may not lie on 4?) are found 
by cutting the surface £ with the equilateral cone of revolution 
with vertical axis, whose vertex lies in P: the circles touching at 
a given circle by cutting 2 with one of the two equilateral cones 
of revolution with vertical axis which have the given circle as base- 
circle; on the other hand we find the circles that cut, besides 47, 
also an arbitrarily given circle perpendicularly, by cutting 2 with 
the equilateral hyperboloid of revolution for which that circle is 
the throat circle. 
Jut instead of the simple figures, point, straight line, and circle, 
a second arbitrary curve #’ may-be considered, of order u’, ete. 
and we may inquire after the circles cutting both these curves 
perpendicularly at a time, especially one twice, the other once; it 
is clear that the answer to any questions that may be put here 
will be obtained by cutting the surfaces 2 and 2’ with each 
other. And if one goes a step farther in this direction and combines 
the surfaces @, 2’, 2’’, all tbe circles that cut 3 given curves at 
a time perpendicularly are found. 
Finally, the cyclographic surfaces, belonging to touching circles 
and perpendicularly cutting circles may be combined together, and 
so e.g. investigate the circles that cut one. of two given curves per- 
pendicularly and touch the other, with the peculiarities consequent 
on this, as e.g. the circles of curvature, or the twice touching 
circles of one curve, which cut the other perpendicularly, or the 
circles that cut one curve twice perpendicularly and touch at the 
other, ete., and if one imagines only one curve as given, but for 
this one constructs both the surface belonging to the touching circles 
and the one belonging to perpendicularly cutting circles, one finds 
by their intersection the circles that touch a given curve and cut it 
perpendicularly at a time, with all the peculiarities that may arise 
here, and without other difficulties having to be overcome with it 
but those comprised in the tracing of the unreal, and therefore to 
be separated, solutions. ; 
