Page:American Journal of Mathematics Vol. 2 (1879).pdf/14

$v_{56}\,(AB\,.DC),$ &c., meet in a single point $Y,$ through which passes also an $i$ line; and all the pairs of points of the same notation, $V_{23},$ $V_{45},$ $V_{67},$ &c., together with the $P$ point of the same notation, lie in a line $y,$ through an $I.$ There are $45$ lines $y,$ three through each $I$ point, and $45$ points $Y,$ three on each $i$ line (p. 52).

The $90$ points $V_{23},$ $V_{45},$ &c., also lie in twos on $180$ lines $n_{23},$ $n_{45},$ &c., respectively, which pass by fours through the $45$ points $Y.$ A similar relation holds between the $v$ lines (p. 60).

Veronese gives many relations of harmonicism and of involution, which I omit. For instance, he shows that the pairs of points $H_{2}H_{3},$ $H_{4}H_{5},$ $H_{6}H_{7},$ &c., of same notation, which lie all on a common $g$ line, form a system of points in involution, whose double points are the $H$ point of same notation and the $I$ point of the $g$ line.

III.

1. Since the point $G\,(ABC\,.DEF)$ is conjugate to the point $G\,(ABC\,.DFE)$ with respect to the conic $S,$ and the pole of the line $g'\,(abc\,.def)$ with respect to the same conic, it follows that the point $G$ is on the line $g'\,(abc\,.def)$ is on the line $g'\,(abc\,.def);$ it is also on the line $g\,(ABC\,.DEF),$ hence it is at their intersection. In general, $g$ lines and $g'$ lines of the same notation intersect in $G$ points. Since in the Brianchon figure the $g'$ lines consist of ten pairs of lines conjugate with respect to $S,$ it may be shown in the same way that $G$ points and $G'$ points of the same notation, as $G\,(AFC\,.BED)$ and $G'\,(afc\,.bed),$ lie on $g'$ lines, as $g'\,(afc\,.bde).$

2. Since $ABCD$ is a quadrilateral inscribed in a conic, the intersections of its diagonals, $P\,(BC\,.AD),$ $P\,(CD\,.AB),$ $P\,(AC\,.BD),$ are the vertices of a triangle self-conjugate to the conic and the line joining $P\,(CD\,.AB)$ to $P\,(AC\,.BD)$ is the polar of $P\,(BC\,.AD);$ but $p'\,(bc\,.ad)$ is also the polar of $P\,(BC\,.AD),$ hence these two lines coïncide. In the same way it may be shown that the point of intersection of $p'\,(cd\,.ab)$ and $p'\,(ac\,.bd)$ coïncides with $P\,(BC\,.AD),$ and, in general, that the triangle whose vertices are the $P$ points obtained from four of the six points on the conic coïncides with the triangle whose sides are the $p'$ lines obtained from the tangents at the same four points. There are $15$ combinations of four letters out of six, hence there are $15$ of these self-conjugate triangles. Since a self-conjugate triangle has