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

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

The connecting link between the system ${\displaystyle [H_{1}h_{1}],}$ and the system ${\displaystyle [H_{2}h_{2}],}$ is formed by the ${\displaystyle 90}$ lines ${\displaystyle v_{12},}$ which fact is indicated by the suffix ${\displaystyle _{12}.}$ We have already seen that the ${\displaystyle v_{12}}$ lines which pass through ${\displaystyle H_{1}\,(BCDEFA)}$ are

Now three ${\displaystyle v_{12}}$ lines which pass through one ${\displaystyle H_{2}}$ point are (Veronese, p. 35)

That is, given three pairs of ${\displaystyle v_{12}}$ lines such that one member of each pair passes through a common ${\displaystyle H_{1}}$ point, the remaining members pass through a common ${\displaystyle H_{2}}$ point. This correspondence between ${\displaystyle H_{1}}$ points and ${\displaystyle H_{2}}$ points I shall indicate by giving two such points the same notation. It will then be observed that the three ${\displaystyle v_{12}}$ lines of one ${\displaystyle H_{1}}$ point are obtained by taking its opposite pairs of letters in the order in which they stand; but the three ${\displaystyle v_{12}}$ lines of one ${\displaystyle H_{2}}$ point by taking opposite pairs of letters with an inversion of one pair. On a ${\displaystyle v_{12}}$ line, ${\displaystyle v_{12}\,(AB\,.CD),}$ lie two ${\displaystyle H_{1}}$ points, ${\displaystyle H_{1}\,(ABECDF),}$ ${\displaystyle H_{1}\,(ABFCDE),}$ and two ${\displaystyle H_{2}}$ points, ${\displaystyle H_{2}\,(ABEDCF),}$ ${\displaystyle H_{2}\,(ABFDCE).}$

The three ${\displaystyle H_{2}}$ points which have the same notation as the three ${\displaystyle h_{1}}$ lines of an ${\displaystyle H_{1}}$ point lie on an ${\displaystyle h_{2}}$ line (Veronese, p. 39). Through each ${\displaystyle H_{2}}$ point pass three ${\displaystyle h_{2}.}$ There are ${\displaystyle 60}$ ${\displaystyle H_{2}}$ points and ${\displaystyle 60}$ ${\displaystyle h_{2}.}$

Two lines ${\displaystyle h_{2}}$ of the same notation as the two ${\displaystyle H_{2}}$ points of one ${\displaystyle v_{12}}$ line meet in a point ${\displaystyle V_{23},}$ through which pass two ${\displaystyle h_{3}}$ lines of the third system ${\displaystyle [H_{3}h_{3}].}$ These ${\displaystyle h_{3}}$ lines, ${\displaystyle 60}$ in number, determine by their intersections in threes the ${\displaystyle 60}$ ${\displaystyle H_{3}}$ points, which lie in threes on the ${\displaystyle h_{3}}$ lines. There are ${\displaystyle 45}$ pairs of points ${\displaystyle V_{23},}$ answering to the ${\displaystyle 45}$ pairs ${\displaystyle P}$ of the system ${\displaystyle [H_{1}h_{1}];}$ that is to say, after the first system the intrinsic difference between ${\displaystyle H}$ points and ${\displaystyle h}$ drops out, or ${\displaystyle h}$ lines no longer meet by fours in ${\displaystyle 45}$ points, but by twos in ${\displaystyle 90}$ points.

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

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