Ladd, The Pascal Hexagram.
11
in this hexagon, 
is the reciprocal of the conic 
with respect to a third conic; 
twelve points of which may be obtained by taking on each side of the Brianchon hexagon the two points which form a harmonic range with each of the two pairs of vertices on this side; for instance, on 
the two points which are harmonic at once with 
and with 
The hexagon 
is the reciprocal with respect to the conic 
of the hexagon formed by joining its alternate vertices; the point 
is the pole of the line 
the point 
is the pole of of the line 
hence the Pascal 
is the polar of the point 
is the pole of 
is the pole of 
hence the Pascal 
is the polar of the point 
It follows that the intersection of the Pascals 
which is the Kirkman 
is the pole of a line joining 
to 
which is the Pascal 
But the six hexagons, 
form, by connectors of alternate vertices, a Brianchon hexagon composed of the same sides in different orders, and hence circumscribed to the same conic, therefore the six Pascals 
are the poles of the six Kirkmans 
with respect to the same conic 
Moreover, the points 
and 
in which the first three and the second three Pascals intersect are the poles respectively of the lines 
and 
which connect the first three and the second three Kirkmans. The two 
points in question are harmonic conjugates with respect to the conic 
hence their polars with respect to the 
lines of the same notation, are harmonic conjugates with respect to the reciprocal conic, 
The triangle whose vertices are two corresponding points and the intersection of the lines through them (or, what is the same thing, the triangle whose sides are two corresponding lines and the line joining the points on them) is a triangle self-conjugate with respect to the conic two of its vertices being at the same time conjugate with respect to and two of its sides with respect to Since this conic, is inscribed in the triangles 
and 
we shall call it the conic 
(where the order of the letters in each group of three is of no consequence) and the conic with respect to which it is the reciprocal of we shall call 
There are ten conics 
the
