Page:American Journal of Mathematics Vol. 2 (1879).pdf/17

Rh in this hexagon, $$\Sigma_1 ,$$ is the reciprocal of the conic $$S$$ with respect to a third conic; $$X_1,$$ twelve points of which may be obtained by taking on each side of the Brianchon hexagon the two points which form a harmonic range with each of the two pairs of vertices on this side; for instance, on $$AC$$ the two points which are harmonic at once with $$C,\,P\,(BD\,.AC),$$ and with $$A,\,P\,(BF\,.AC).$$ The hexagon $$ABCDEF$$ is the reciprocal with respect to the conic $$X$$ of the hexagon formed by joining its alternate vertices; the point $$P\,(BD\,.AC)$$ is the pole of the line $$BC,$$ the point $$P\,(AE\,.FD)$$ is the pole of of [sic] the line $$FE,$$ hence the Pascal $$h\,(CAEBDF)$$ is the polar of the point $$P\,(BC\,.EF);$$ $$P\,(BD\,.CE)$$ is the pole of $$CD,\,P\,(FB\,.AE)$$ is the pole of $$AF,$$ hence the Pascal $$h\,(AECFBD)$$ is the polar of the point $$P\,(CD\,.AF).$$ It follows that the intersection of the Pascals $$h\,(CAEBDF),$$ $$h\,(AECFDB),$$ which is the Kirkman $$H\,(AFEDCB),$$ is the pole of a line joining $$P\,(CD\,.AF)$$ to $$P\,(BC\,.FE),$$ which is the Pascal $$h\,(AFEDCB).$$ But the six hexagons, $$ABCDEF,$$ $$AFCBED,$$ $$ADCFEB,$$ $$ABCFED,$$ $$ADCBEF,$$ $$AFCDEB,$$ form, by connectors of alternate vertices, a Brianchon hexagon composed of the same sides in different orders, and hence circumscribed to the same conic, therefore the six Pascals $$h\,(ABCDEF),$$ $$h\,(AFCBED),$$ $$h\,(ADCFEB),$$ $$h\,(ABCFED),$$ $$h\,(ABDCEF),$$ $$h\,(AFCDEB),$$ are the poles of the six Kirkmans $$H\,(ABCDEF),$$ $$H\,(AFCBED),$$ $$H\,(ADCFEB),$$ $$H\,(ABCFED),$$ $$H\,(ADCBEF),$$ $$H\,(AFCDEB),$$ with respect to the same conic $$X_1.$$ Moreover, the points $$G\,(ACE\,.BDF)$$ and $$G\,(ACE\,.BFD)$$ in which the first three and the second three Pascals intersect are the poles respectively of the lines $$g\,(ACE\,.BDF)$$ and $$g\,(ACE\,.BFD)$$ which connect the first three and the second three Kirkmans. The two $$G$$ points in question are harmonic conjugates with respect to the conic $$S,$$ hence their polars with respect to $$X_1,$$ the $$g$$ lines of the same notation, are harmonic conjugates with respect to the reciprocal conic, $$\Sigma_1.$$ The triangle whose vertices are two corresponding $$G$$ points and the intersection of the $$g$$ lines through them (or, what is the same thing, the triangle whose sides are two corresponding $$g$$ lines and the line joining the $$G$$ points on them) is a triangle self-conjugate with respect to the conic $$X_1,$$ two of its vertices being at the same time conjugate with respect to $$S,$$ and two of its sides with respect to $$\Sigma_1.$$ Since this conic, $$\Sigma_1,$$ is inscribed in the triangles $$ACE$$ and $$BDF,$$ we shall call it the conic $$\Sigma \,(ACE\,.BDF),$$ (where the order of the letters in each group of three is of no consequence) and the conic with respect to which it is the reciprocal of $$S$$ we shall call $$X\,(ACE\,.BDF).$$ There are ten conics $$\Sigma,$$ the