Feb., 1916] Hexagon Notation 143 



(31) To write four g-points on an I-line. 

 Use (30), which shows there are 15 I-Hnes. 



(32) The four g-points of an I-line are on the Pascals of the 

 hexagotis unique to the four h-points of the corresponding h-point 

 quadrangles as given in {27); as also on the Pascals of the hexagons 

 ordinary. 



The hexagons unique to the four h-points in (27) are: 



fadbf ce (1)1 

 J a b c f e d (2) 



lacef db (3)f ^^^• 



[ae df b c (4^ 

 and (1) is in g4; (2) in gi; (3) in gs; (4) in g4. 



[Note that hne (2) of (A) is formed from Hne (1), by writing 

 first the alternate letters of (1), abc; then the second set of 

 alternate letters of (1), beginning with the fourth letter from 

 the initial letter of the first set, the second set being taken in 

 the same direction as the first set. The set is unique]. 



By (11) the same g-points fixed by (A) are also on the 

 Pascals of the hexagons ordinary, one set of which for the 

 h-points of (27) is: 



af, be, cd (I) ] 



af, cd, eb (II) 



af, eb, dc (III) [ ^^^• 

 af, dc, be (IV) 



(33) Set (A) of (32) shows there are 15 I-lines, since any 

 line of (A) uniquely determines all the rest. There are thus 

 only 15 such complete quadrilaterals as (A). 



In (A) there are 4 Pascal lines meeting in six Pascal points, 

 and the triangle of any three of the lines is the Pascal triangle 

 (7) of the fourth hexagon. These fifteen quadrilaterals deter- 

 mine the most important features of the hexagon geometry. 

 (See second paper.) 



(34) To write three I-lines through any g-point. 



Form from each line of the g-point a group like (A) of (32), 

 for the initial lines of the four g-points on each of the three 

 I-lines. Therefore through each g-point pass three I-lines of 

 four g-points each. 



For the g-point 



a b c d e f 



a f c b e d}' (gi). 



a d c f e b 



