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

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

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

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

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

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