Page:American Journal of Mathematics Vol. 2 (1879).pdf/18

reciprocals of $S$ with respect to ten conics $X.$ Each pair of corresponding $G$ points and the six Pascals through them are reciprocal, with respect to one conic $X,$ to the two $g$ lines and the six $H$ points of the same notation. The $60$ $H$ points and the $60$ $h$ lines are then divided into ten systems of six lines and points each, reciprocal to each other with respect to the $X$ conic of that system.

These properties of the Pascal Hexagram can be summed up in the following propositions:

(1). The ${\textit {20}}$ Steiner points $G$ are the intersections of the ${\textit {20}}$ Cayley-Salmon lines $g$ with the ${\textit {20}}$ corresponding lines $g'.$ The ${\textit {20}}$ lines $g'$ are the connectors of the ${\textit {20}}$ Steiner points $G$ with the ${\textit {20}}$ corresponding points $G'.$

(2). The ${\textit {45}}$ points $P$ lie in twos on ${\textit {45}}$ lines $p'{\textit {,}}$ which meet by threes in ${\textit {60}}$ points $H'{\textit {,}}$ the poles with respect to the original conic of the $h$ lines. The $H'$ points lie in fours on the lines $p'{\textit {,}}$ in threes on the ${\textit {60}}$ lines $h'$ and in threes on ${\textit {20}}$ lines $g'.$ From them may be produced any number of systems of points and lines, $[H'_{n}h'_{n}]{\textit {,}}$ having their $g'$ and $i'$ lines and their $G'$ and $I'$ points in common. But in this case transition is made from a system of even index to one of odd by any means of $v'$ lines, and from one of odd to one of even by means of $V'$ points.

(3). Three Pascal lines which belong to a triangle formed of fundamental sides intersect those sides in a $k$ line. There are ${\textit {60}}$ lines $k.$ Their intersections with corresponding $h$ lines lie in fours on ${\textit {15}}$ lines $l.$

(4). Of the corresponding circumscribed and inscribed triangles of the conic, the ${\textit {20}}$ centres of homology, $C{\textit {,}}$ lie in twos on ${\textit {90}}$ lines $c{\textit {,}}$ which pass by twos through the ${\textit {45}}$ points $P{\textit {,}}$ and the ${\textit {20}}$ axes of homology, $a{\textit {,}}$ intersect in twos in ${\textit {90}}$ points $A{\textit {,}}$ which lie in twos on the ${\textit {45}}$ lines $p'.$

(5). The $H$ points and the $h$ lines may be divided into ten groups of six lines and points each. The lines and points of each group are poles and polars with respect to one of ten auxiliary conics $X.$ To each group belong two corresponding $G$ points and two corresponding $g$ lines. They form a triangle self-conjugate with respect to the $X$ conic of the group. The $G$ points are at the same time conjugate with respect to the conic $S{\textit {,}}$ and the $g$ lines are at the same time conjugate with respect to the conic, $\Sigma {\textit {,}}$ the $X$ reciprocal of $S.$