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

8 $$v_{56}\,(AB\,.DC),$$ &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 $$H_2H_3,$$ $$H_4H_5,$$ $$H_6H_7,$$ &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.

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