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



In (g') there is also cyclic permutation of the initial letters 

 of the segments in one direction (to the right) and of the final 

 letters in the other direction (to the left). This offers the 

 easiest way of writing (g')- 



In (h'), line (2) is formed from line (1) by setting under 

 each segment of (1) its opposite segment, retaining the hexagon 

 order of letters in {!) in one column (here the first) and reversing 

 it in- the other two; (3) from (2) as (2) from (1); (1') from (3) 

 as (3) from (2); (1') is (1). The lines (1) and (2) give the 

 Pascal of (1); (2), (3) that of (2); (3), (1') that of (3). 



(3) When only the hexagons which enter into (g) and 

 (h) are desired, they may be written thus : 



'a b c d e f ) (1) fc e a, b f d] (1) 



af cbed (2) (g"); ]eac, dbf^ (2) (h") 



^a d c f e bj (3) [a c e, f d bj (3) 



In (g") one set of alternate letters (here a, c, e) is held 

 fixed; the other set permuted cyclically (in either direction). 



In (h"), line (1) is divided into two groups by a comma. 

 Each group is permuted cyclically, the one in the opposite 

 direction of the other. If the hexagons in (h") are to indicate 

 the same point as (h'), set astride the comma the segment of 

 the first line of (h) in the column which was to hold the hexagon 

 order (here ab) in (h")- 



These notations lend themselves, as will be seen in the 

 following, most readily to determine the whole geometry of the 

 hexagon configuration. 



(4) The g-point of any hexagon is the center of perspective 

 of its two triangles of alternate sides, the axis of perspective 

 being the Pascal line of the hexagon. 



To write the g-point of any hexagon, interchange any pair 

 of its alternate letters for the first line and proceed as in (g') or 



For a b c d e f (interchanging c and e) it is: 



fab, ed, cfl fa b e d c f' 



\ cd, af, eb > (gO or <^afebcd^ (g2). 



[ef, cb, adj [a d e f c b^ 



(5) The conjugate g-point of any given g-point 



The g-point of any hexagon of a g-point is one and the same 

 g-point (the conjugate g-point). 



