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k» twice perpendicularly, the locus therefore of the points out of 
which two equally long tangents may be drawn at 4”. If the 
calculation is carried out, we find: 
The locus of the points out of which two equally long tangents 
may be drawn at hk” is a curve of order: 
de —!(4uy + wv? + 5u — 13p + 31 — 8ve — 2v6 — 36? + Be + 50). 
And according to the preceding observations this curve has in 
each node of 4” 2 cusps, while through each cusp of 4 pass 3 
branches, which all three touch at the euspidal tangent. Through 
each vertex of 4” and through each focus the curve passes once. 
For the hyperbola and the ellipse we find; 
de = 1 (4.2.2 +. 2? + 5.2 —13.2).= 2, viz. the two axes; for the 
parabola: 4 (4.2.2 + 2? + 5.2 — 13.2 — 2.2.1 — 3.1° + 5.1) =—1, viz. 
the axis. 
We may observe moreover that the curve found here is of course 
only partly active, and for the rest parasitic; the parasitic parts are, 
however, of two kinds: some parts of the curve are centres of 
circles with imaginary radius, others on the other hand of real 
circles, which, however, do not cut 4” perpendicularly in a real 
way, i.e. where exactly those points, where the intersection takes 
place perpendicularly, are imaginary. So as to the ellipse the parts 
of the major axis lying outside the ellipse, are active (cf. § 5), the 
parts between the vertices and the foci are centres of imaginary 
circles, whereas the part between the two foci contains the centres 
of real circles, which, however, do not cut /’ perpendicularly in a 
real way. As the branches of the nodal curve which pass through 
the foci extend to either side of # as far as point Z,, radii of any 
greatness must be found in the eyclographic representation of those 
branches, from the zero circle, which corresponds with the focus, 
to the straight line at infinity, which represents Z, cyclographically. 
The circles representing the points of the nodal curve in the close 
neighbourhood of the focus are very small and lie therefore entirely 
within the ellipse; but there are also very great circles, and so there 
must be a circle that meets the ellipse really for the first time. This 
meeting must of course be contact, and this contact will take place in 
the vertex nearest to the focus; the circle then touches at the ellipse in 
its vertex and cuts it perpendicularly in two imaginary points. The 
two intersections coinciding in the vertex diverge now, describe the 
ellipse, meet again in the other vertex, and after that the circle 
will enclose the ellipse entirely, 
