Page:American Journal of Mathematics Vol. 2 (1879).pdf/12

6 connected two and two by $$90$$ lines $$v,$$ which pass each through two points $$P.$$ Professor Cayley and Dr. Salmon discovered at the same time that $$60$$ Kirkman points lie in threes on $$20$$ (Cayley-Salmon) lines $$g,$$ and Dr. Salmon, that the lines $$g$$ meet in threes in the $$15$$ (Salmon) points $$I.$$ Hesse pointed out the correspondence which exists between the lines and points of the figure; but he was aware that the relation is not that of pole and polar, at least not with respect to the original conic. (Crelle, Vol. 68, p. 193.)

Veronese has written a paper (Nuovi Teoremi sul Hexagrammum Mysticum, Reale Accademia dei Lincei, 1876-1877,) which apparently leaves little work for other investigators to do. His most important discovery is that the $$60$$ $$h$$ lines may be divided into six groups of ten lines each, which intersect in the ten corresponding $$H$$ points are are their polars with respect to a conic $$\pi.$$ There are six conic $$\pi$$ in the whole figure, and any five of these groups of ten lines and points determine the sixth. He has shown, moreover, that besides the original system, $$[H_1h_1],$$ of $$60$$ Pascal lines and Kirkman points, there is an infinity of such systems, $$[H_nh_n],$$ consisting each of six groups of ten lines and points, and giving rise each to six conics. Five groups of any system after the first suffice to determine one group of the preceding and one of the succeeding system. The figure of the $$g$$ lines and of the $$G$$ points is common to all these systems; that is to say, the $$60$$ $$H$$ points of every system lie in threes on the same $$20$$ $$g$$ lines and the $$60$$ $$h$$ lines of every system pass by threes through the same $$20$$ $$G$$ points. It follows that the $$I$$ points and the $$i$$ lines are also common to all the systems. Veronese uses the symbol $$\pi$$ for a group of ten lines and points as well as for the conic with respect to which they are poles and polars. He gives a table by consulting which one can see to what figure $$\pi$$ any $$h$$ line belongs. But the $$h$$ lines which go together to form a figure $$\pi$$ can be determined at once by observing the following rule: Take any $$h$$ line, the other six $$h$$ lines through the three $$H$$ on it, and the three $$h$$ lines through the $$H$$ point which corresponds to it; these ten $$h$$ lines constitute a figure $$\pi,$$ to which belong also the ten $$H$$ points of the same notation. A symbol for a figure $$\pi$$ thus obtained, from which symbol it can be known immediately whether a given line or point belongs to the figure which it represents or not, is a desideratum. Veronese calls his figures $$\pi$$ first, second, third, &c., and the connection between the first figure and its lines and points is of course entirely arbitrary. No two $$h$$ lines of one figure $$\pi$$ pass through a common $$G$$ point, hence to a figure $$\pi$$ correspond ten differ-