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



Therefore each hexagon enters into only one g-point. Therefore 

 twenty g-points. 



(15) The number of h-points, and how to write three h-points 

 on any Pascal line. 



In forming (h') of section (2) we held the hexagon order in 

 one column and reversed in two. This can be done in three 

 ways, therefore each hexagon enters into three h-points, there- 

 fore, three times as many h-points as g-points. Therefore 

 sixty h-points. 



This is also shown in the notation of section (3), since 

 the line (1) can be grouped in three ways as indicated. 



To write three h-points on a Pascal line, proceed as in h' of 

 (2), retaining the hexagon order in columns 1, 2, 3, in order;, 

 or, if using the notation in (3), use as initial lines, abc, def ; bed, 

 efa; cde, fab. 



(16) The number of G-lines. 



By (13) each g-point gives a G-line through its conjugate 

 g-point. 



Therefore, twenty G-lines. 



(17) Given a g-poirit to write the three h-points on a G-line 

 with it. 



Write the conjugate g-point (5) and the three h-points 

 unique to its hexagons ( (9), (13) ). Also on any Pascal line 

 there is one g-point and three h-points, ( (14, (15) ). 



(18) Given one h-point of a G-line to write the g-point and two 

 remaining h-points. 



Write the hexagon unique (10) to the given h-point; then the 

 two hexagons which enter with this hexagon into a g-point 

 ( (2) or (3) ) ; then their two unique h-points, and the conjugate 

 g-point. Thus, through any h-point goes only one g-line. 



(19) To write three g-points on a line. 



Write the g-points of the hexagons of any h-point. The 

 triangles of alternate sides of the hexagons of an h-point are 

 but three in perspective in pairs at three g-points (4), with axes 

 of perspective concurrent in the h-point; therefore the three 

 centers of perspective are collinear. 



(20) To write three lines of three g-points each, with a grpoint 

 in common. 



Write the three lines of g-points of three h-points on any 

 Pascal line, (15). The g-point in common to the three lines is 

 the g-point conjugate to that on the common Pascal line. 

 ( (19) and (5) ). 



