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

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

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

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 $$59$$ other Pascal lines. By attending to the notation of Pascal lines such a table may be dispensed with. His $$90$$ points, "$$m,$$" $$360$$ points "$$r,$$" 360 points "$$t,$$" $$360$$ points "$$z,$$" and $$90$$ points "$$w$$" are the intersections each of two Pascals whose symbols can easily be derived one from another. For instance,

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 $$H',\,G',\,I'$$ points, which are the poles of the $$h,\,g,\,i$$ lines, and for the $$h',\,g',\,i'$$ lines, which are the poles of the $$H,\,G,\,I$$ points.

It was shown by Kirkman that the two Kirkman points

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

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