144 



The Ohio Journal of Science [Vol. XVI, No. 4, 



The three hnes of g-points are (by initial lines) 



'a b c d e f (gi)l fa f c b e d (gi)j fa d c f e b (gi) 



a c e d f b (gs) I 1 a c e b d f (g5) I I a c e f b d (gg) 



a e f d b c (ga) [ ' | a e d b f c (ge) ( ' ) a e b f d c (gg) 



[a f b d c e (g4)J [a d f b c e (gr)] [a b d f c e (gio; 



In a set of g-points like the ten above no two are conjugate 

 to each other. 



(35) To write two sets of conjugate g-points, of ten points in 

 each set, each set having a point in it conjugate to a point in the 

 other set. 



Treat two conjugate g-points as was (gi) of (34). 



(36) To write four G-lines concurrejit in an i-point. (Sal- 

 mon's notation). 



Write four g-points on a line (by (31) or (32) ) ; then the 

 three h-points unique to the hexagons of each of the collinear 

 g-points. This will give four lines containing three h-points 

 each, one line passing through each of the g-points conjugate 

 to the four collinear g-points. 



These four g-lines are concurrent in an i-point. 



The triangles of alternate sides of the hexagons of a g-point 

 are but three (A, B, C). Their Pascal triangles (7) are three 

 (Pi, P2, P3). By (4) A, B, C are in perspective at the conjugate 

 g-point (gi) ; by 7 and 8, A, Pi, Po are in perspective at hi; 

 B, P2, P3 at h2; C, P3, Pi at h.3. Therefore, g' , hi, ho, hs are 

 collinear (a G-line). It follows easily that for four collinear 

 g-points, the resulting four G-lines are concurrent, 



(37) There are three i-points on each G-line. 



Through each g-point there are three I-lines of four g-points 

 each (34). Each gives an i-point on the G-line through the 

 conjugate g-point, as in (3G). 



(38) The three h-points unique to the hexagons of an h-point 

 are all on the Pascal line of the hexagon unique to the given h-point. 



The h-point unique to a b c d e f (1) is, by (9), 



face, bfd (2)^ 

 cea, dbf (3n (hi). 

 ^eac, fdb (4)^ 



