Page:American Journal of Mathematics Vol. 2 (1879).pdf/13

Rh ent $$G$$ points. No two of them are conjugate $$G$$ points. Any two figures $$\pi$$ have in common four $$G$$ points which lie on one $$i$$ line, or to each $$i$$ line corresponds one of the $$15$$ possible combinations two by two of the six figures $$\pi;$$ and the four $$g$$ lines common to any two figures $$\pi$$ pass through an $$I$$ point.

The connecting link between the system $$[H_1h_1],$$ and the system $$[H_2h_2],$$ is formed by the $$90$$ lines $$v_{12},$$ which fact is indicated by the suffix $$_{12}.$$ We have already seen that the $$v_{12}$$ lines which pass through $$H_1\,(BCDEFA)$$ are

Now three $$v_{12}$$ lines which pass through one $$H_2$$ point are (Veronese, p. 35)

That is, given three pairs of $$v_{12}$$ lines such that one member of each pair passes through a common $$H_1$$ point, the remaining members pass through a common $$H_2$$ point. This correspondence between $$H_1$$ points and $$H_2$$ points I shall indicate by giving two such points the same notation. It will then be observed that the three $$v_{12}$$ lines of one $$H_1$$ point are obtained by taking its opposite pairs of letters in the order in which they stand; but the three $$v_{12}$$ lines of one $$H_2$$ point by taking opposite pairs of letters with an inversion of one pair. On a $$v_{12}$$ line, $$v_{12}\,(AB\,.CD),$$ lie two $$H_1$$ points, $$H_1\,(ABECDF),$$ $$H_1\,(ABFCDE),$$ and two $$H_2$$ points, $$H_2\,(ABEDCF),$$ $$H_2\,(ABFDCE).$$

The three $$H_2$$ points which have the same notation as the three $$h_1$$ lines of an $$H_1$$ point lie on an $$h_2$$ line (Veronese, p. 39). Through each $$H_2$$ point pass three $$h_2.$$ There are $$60$$ $$H_2$$ points and $$60$$ $$h_2.$$

Two lines $$h_2$$ of the same notation as the two $$H_2$$ points of one $$v_{12}$$ line meet in a point $$V_{23},$$ through which pass two $$h_3$$ lines of the third system $$[H_3h_3].$$ These $$h_3$$ lines, $$60$$ in number, determine by their intersections in threes the $$60$$ $$H_3$$ points, which lie in threes on the $$h_3$$ lines. There are $$45$$ pairs of points $$V_{23},$$ answering to the $$45$$ pairs $$P$$ of the system $$[H_1h_1];$$ that is to say, after the first system the intrinsic difference between $$H$$ points and $$h$$ drops out, or $$h$$ lines no longer meet by fours in $$45$$ points, but by twos in $$90$$ points.

In general, from the system $$[H_{2n-1}h_{2n-1}]$$ the system $$[H_{2n}h_{2n}]$$ is derived by means of lines $$v_{2n-1,\,2n},$$ the connectors of pairs of $$H_{2n-1}$$ points and also of pairs of $$H_{2n}$$ points. From the system $$[H_{2n}h_{2n}]$$ we pass to the system $$[H_{2n+1}h_{2n+1}]$$ by means of points $$V_{2n,\,2n+1},$$ the intersections of pairs of $$h_{2n}$$ lines and also pairs of $$h_{2n+1}$$ lines.

All the pairs of $$v$$ lines of same notation but from different systems, $$v_{12}\,(AB\,.CD),$$ $$v_{12}\,(AB\,.DC);$$ $$v_{34}\,(AB\,.CD),$$ $$v_{34}\,(AB\,.DC);$$ $$v_{56}\,(AB\,.CD),$$