Page:American Journal of Mathematics Vol. 2 (1879).pdf/9

Salmon line, the line ${\displaystyle g\,(ACE\,.BFD).}$ It passes through the point ${\displaystyle G\,(ACE\,.BDF).}$ Through the two conjugate points ${\displaystyle G\,(ACE\,.BDF),}$ ${\displaystyle G\,(ACE\,.BFD),}$ pass respectively the lines ${\displaystyle g\,(ACE\,.BFD),}$ ${\displaystyle g\,(ACE\,.BDF).}$ These two ${\displaystyle g}$ lines we shall call, for the present, corresponding ${\displaystyle g}$ lines. They are not conjugate with respect to the conic ${\displaystyle S.}$ (Veronese, Nuovi Teoremi sul Hexagrammum Mysticum, p. 26.) The ${\displaystyle H}$ points on ${\displaystyle G\,(ACE\,.BDF)}$ correspond to the ${\displaystyle h}$ lines through ${\displaystyle g\,(ACE\,.BDF);}$ hence we shall say that ${\displaystyle g\,(ACE\,.BDF)}$ corresponds to ${\displaystyle G\,(ACE\,.BDF),}$ while it passes through ${\displaystyle G\,(ACE\,.BFD).}$

The symbol for the Salmon point in which four ${\displaystyle g}$ lines intersect is obtained in the same way as that of the Steiner-Plücker line through four ${\displaystyle G}$ points. In fact, the lines ${\displaystyle g\,(BDA\,.ECF),}$ ${\displaystyle g\,(EDF\,.BCA),}$ ${\displaystyle g\,(BCF\,.EDA),}$ ${\displaystyle g\,(BDF\,.ECA),}$ intersect in the Salmon point ${\displaystyle I\,(BE\,.CD\,.AF);}$ and the ${\displaystyle I}$ points on ${\displaystyle g\,(ACE\,.BDF),}$ are ${\displaystyle I\,(AB\,.CD\,.EF),}$ ${\displaystyle I\,(AD\,.CF\,.EB),}$ ${\displaystyle I\,(AF\,.CB\,.ED).}$

Professor Cayley (Quarterly Journal, Vol. IX,) gives a table to show in what kind of a point each Pascal line meets every one of the ${\displaystyle 59}$ other Pascal lines. By attending to the notation of Pascal lines such a table may be dispensed with. His ${\displaystyle 90}$ points, "${\displaystyle m,}$" ${\displaystyle 360}$ points "${\displaystyle r,}$" 360 points "${\displaystyle t,}$" ${\displaystyle 360}$ points "${\displaystyle z,}$" and ${\displaystyle 90}$ points "${\displaystyle w}$" are the intersections each of two Pascals whose symbols can easily be derived one from another. For instance,

${\displaystyle {h\,(ABCDEF)} \atop {h\,(ABCFED)}}$ ${\displaystyle >}$ "${\displaystyle m,}$"	${\displaystyle {h\,(DEFABC)} \atop {h\,(DEFBCA)}}$ ${\displaystyle >}$ "${\displaystyle r,}$"	${\displaystyle {h\,(ACEDBF)} \atop {h\,(ABCDEF)}}$ ${\displaystyle >}$ "${\displaystyle t,}$"
${\displaystyle {h\,(ACFBED)} \atop {h\,(AFEDCB)}}$ ${\displaystyle >}$ "${\displaystyle z,}$"		${\displaystyle {h\,(ACEBFD)} \atop {h\,(ABCDEF)}}$ ${\displaystyle >}$ "${\displaystyle w.}$"

By producing the lines and points of the Brianchon hexagon, as we may call the corresponding circumscribed hexagon, we should find occasion for the same symbols, in small letters, for the ${\displaystyle H',\,G',\,I'}$ points, which are the poles of the ${\displaystyle h,\,g,\,i}$ lines, and for the ${\displaystyle h',\,g',\,i'}$ lines, which are the poles of the ${\displaystyle H,\,G,\,I}$ points.

It was shown by Kirkman that the two Kirkman points

are on a line through the point ${\displaystyle P\,(AB\,.FE).}$ I shall call this line ${\displaystyle v_{12}\,(BF\,.EA)}$ (and it happens that my notation here coïncides with that of Veronese, p. 43). So the points

are on the line ${\displaystyle v_{12}\,(BF\,.AE),}$ which passes through ${\displaystyle P\,(EB\,.FA)}$ and which does not coïncide with ${\displaystyle v_{12}\,(BF\,.EA).}$ Through each point ${\displaystyle P}$ pass two ${\displaystyle v_{12}}$ lines,