Feb., 1916] Hexagon Notation 141 



(21) The nine h-points on three Pascal lines which meet in 

 a g-point establish the same three lines noted in {20). (See (11) ). 



If the g-point lines in (20) are, 



fe ' §^> S3 

 & ' &'4» &5 

 S > &6> &7 



those noted in (21) will be (in addition), easily tested by writing 

 them in full: 



g', g2, gs; 



cr' cr, cr^- 



g', g6, gio; 



g', g3, gs; 



g', g5, gg; 



g', g7, gio; 



which are the same lines as in (20), since two points fix a line. 

 Here g' is the conjugate of the g-point of the three given Pascal 

 lines. 



(22) By (21) the nine h-points on three Pascal lines meeting 

 at a g-point establish three lines of four g-points each through 

 the conjugate g-poi?it. (Salmon's I-lines). 



These lines are established by difi'erent sets of three points, 

 the conjugate g-point always included. 



(23) Through the g-point where three Pascal lines meet goes 

 also one G-line {17), with three h-points. These three h-points 

 establish the same I-lines noted in {22), by sets of three points, the 

 conjugate g-point always excluded. 



The lines will be (if written out) : 



g2, g3, gs; 

 g4, go, gg; 

 gs, g7, gio. 



(24) By (22 and (23) it follows that the twelve h-points 

 of the four lines of h-points (three Pascal lines and one G-line) 

 passing through a g-point establish three I-lines of four g-points 

 each through the conjugate g-point, 



(25) By (24) there are fifteen I-Hnes. 



(26) By (24) there must be four h-points grouped about 

 a g-point (one on each of its Pascal lines and one on its G-line) 

 which establish a single line of four g-points through the con- 

 jugate g-point; and for each g-point three such h-point 

 quadrangles. 



