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conies having & as chord of oseulation BS. As Bis vertex of 
three 8? consequently coincides thrice with JS, the locus of S will 
be a curve o°, having a triple point in 4 and nodes in the remaining 
base-points. It passes moreover through the cyclic points /, 7, on 
/,, for on the @? laid through / that point S belongs to all circles 
of osculation. 
We can now easily point out the five cireles y, passing through 
two base-points B, 2,: two osculate in 4, and intersect in D,, 
two osculate in B, and intersect in B,, the fifth consists of BB, 
and: We: 
That any line / is chord of osculation for five circles y, may be 
proved as follows. 
If in each point L of / the tangent ¢ is drawn at the 98° passing 
through ZL, a system of rays with index 3 is obtained; for / is 
touched by two 98°, is consequently bitangent of the curve enveloped 
by ¢. Through £ we draw the lines w and w’ parallel to the axes 
of B? and the lines » and v’, bisecting the angles between / and t. 
If the pair of lines v,v’ coincides with w,u’, Lis chord of oscu- 
lation of £?. 
If t,u,u’,v,v’, retaining their directions, are transferred to a point 
O of /, a correspondence (4,6) arises in the pencil of rays (U). 
For a ray u determines ($ 1) one 6’, so two points £ and four 
rays v; a ray v determines three tangents f, therefore six rays u. 
The ten coincidences wv form five orthogonal pairs; so there are 
Jive conics for which Lis chord of osculation. 
If the point of contact L of a y, describes the straight line /, the 
end S of the chord of osculation will describe a curve of order 
thirteen, for on / lie eight vertices of conics. 
7. On each conic @ arises a cubic involution, if the three points 
R, of which the osculating circles meet in a point S of 9’, are 
joined into a group. 
If 6’ is an hyperbola, this /, has the points of contact of the 
asymptotes as triple elements; these two replace the four groups 
with a two-fold element, which an /, possesses in general. 
For the ellipse these triple elements become imaginary ; for if it 
is considered as the orthogonal projection of a circle, the /, appears 
to be the projection of the /, formed by the angular points of the 
regular triangles described in that circle. 
For a parabola each group of the /, consists of a point of the 
parabola and the point at infinity of that curve counted twice. 
Let us now consider the triple involution T,, in the plane r, 
