256 
a cusp in the intersection of the inflectional tangent; if therefore 
the curve is symmetrical with regard to >, and Z, a point of 
inflection, the projection consists of a branch ending in the inter- 
section of the asymptotes in Si, and S.,, and which is described 
to and fro. This arises, however, 4 times, and the 4 branches ending 
thus in one and the same point run together into a curve with a 
node, for which the discontinuity is again cancelled. And if we have 
to do with branches which are each other’s images and for which 
Z,, is consequently an ordinary point, these branches project them- 
selves in pairs in one and the same branch, which passes, however, 
through the intersection of the asymptotes of 4”, and in this way a 
node also arises in that case. So we have for all cases the following 
proposition: the locus of the points out of which two equally long tangents 
may be drawn at k* has nodes in the } (u—2e—2o) (u—2e—2o0_1) 
intersections of the asymptotes of ht. 
For the general conic this phenomenon arises once: in fact the 
locus in question consists here of the two axes, and therefore has 
a node in the centre; the 4 branches passing through Z, go to the 
foci here. 
If two asymptotes of i” are chosen arbitrarily, and a hyperbola 
is constructed, which has these two lines for asymptotes, and e.g. 
in order to arrive at the greatest possible contact with /”, this curve 
is osculated in one of the two points at infinity in question, the 
difference between the tangents at the hyperbola and at the curve 
becomes practically imperceptible for some point or other in the 
immediate neighbourhood of the intersection of the asymptotes; from 
which we may conclude that our locus of points of equal tangents 
at A” passes the intersection of the two asymptotes in the same 
directions as the axes of the hyperbola, viz. in the directions of the 
bisectrices of the angles of the asymptotes. We may therefore com- 
plete the above found property of our curve by adding that the two 
nodal tangents in an intersection of two asymptotes of kv bisect the 
angles of those asymptotes. 
The branches of the 2"d kind passing through Z, arise from the 
points of contact of 47 and /,, and appear in groups of 8 ata time 
(cf. $ 6); they are in pairs each other’s images with regard to 
8, or, if we subject @ to a projective transformation, they are asso- 
ciated in pairs to each other in the involutory collineation of which 
Z,, is the centre, and # is the plane. 
The 8 branches passing through Z, have therefore in this point 
only 4 tangents (entirely lying in ¢,), and the intersections of. these 
gangents (lying on 1) are simple points of the locus of the points 
