138 The Ohio Joiirjial of Science [Vol. XVI, No. 4, 



(8) The two ordifiary hexagons of any h-point. 



While the hexagons of a g-point give only one g-point, they 

 give, not six h-points, but only three. 



Any two hexagons related as 



ab, cd, ef 

 ba, dc, fe 



will have one of their ordinary h-points in common. This 

 relation is the same as, 



ab, cd, ef 



ab, ef, cd 



The hexagons of any g-point show this sort of relation in 

 pairs. 



For a given h-point, like 



ace, fdb] 

 cea, bfd \ (C) 

 eac, dbfj 



its two ordinary hexagons are given by reading the first and 

 last letters of each line, in regular order, down the lines (or 

 last and first) giving, 



ab, cd, ef 



ba, dc, fe; 



also straddling the comma. 



In the notation like h' of Section (2), the two hexagons 

 are gotten by reading zig-zag, the column which retained the 

 hexagon order. For h', it was the first column, giving ab, 

 cd, ef and ba, de, fe. 



(9) The unique h-poi?ii of any hexagon. 



In any h-point, as (C) in (8), there are only nine of the 

 fifteen hexagon lines. In (C) they are ac, ce, ef, fd, db, ba, 

 ea, bf, cd. 



The remaining six lines form a hexagon related thus uniquely 

 to the given h-point. 



To write the first line of the unique h-point of any hexagon, 

 a b c d e f (1), write the alternate letters in two groups; ace, bfd; 

 the second group begins with the letter adjacent the initial 

 letter of the first group on the side in the direction of the first 

 grouping and is taken in the direction opposite the first. 



