Feb., 1916] 



Hexagon Notation 



149 



(c) Any two points in a horizontal line are in a line with 

 the conjugates of those not in the line nor columns of this 

 selection (2, 5, 9' 10' is a line; 3, 9, b' , 7' is a line, etc.) 



d The conjugates of the horizontal lines are in line with 

 V {V, 2', 5', 8' is a line, etc.) 



Proof of (c) and (b) : 



Collinear groups with (2) (in addition to that given) are 

 (using other hexagons in a g-point with (2) ) : 



fac, eb, df (2)1 faf, ec, db (2) ] 



ac, df, be (5) I /-r \ j af, db, ce (6') 



ac,be,fd (10') f ^^ ' ] af , ce, bd (8) 



^ac, fd, eb(9')J [af, bd, ec (7')J 



as tested by sections 4 and 5. 



(M) 



And collinear groups with (3) are: 



ffc, ae, db (3)1 fde, fb, ac (3) ] 



fc, db, ea (8') 1 .^s I de, ac, bf (5') 



fc, ea, bd (6) f ^^ ' | de, bf, ca (9) 



[fc,bd,ae (100 J [de, ca, fb (7') 



(O) 



(L), (M), (N), (O) prove (C), while (L) and (M) show that 

 line 2, 5 meets line 3, 6 at 10', and so in same way for other 

 statements. 



Taking the g-point conjugate to 1, and treating it the same 

 way as 1, we get 



V 



6' 5' 7' 



3' 2' 4' 



9' 8' 10 

 as is easily tested. 



Thus the fifteen I-lines given by the numerical table are: 



1, 2, 3, 4 



1, 5, 6, 7 



1, 8, 9, 10 



2, 5, 9', 10' 



2, 8, 6', 7' 

 5, 8, 3', 4' 



3, 6, 8', 10' 

 3, 9, 5', 7' 



as illustrated on the following diagram 



6, 9, 2', 4' 

 4, 7, 8', 9' 

 4, 10, 5', 6' 



7, 10, 2', 3' 

 V, 2', 5', 8' 

 1', 3', 6', 9' 

 1', 4', 7', 10' 



