igo4\ Henderson — Graphical Representation. 131 



will be projected into a line ab, and the point of intersection 

 of the planes a, b, c into a point abc; and so in the other cases 

 (recalling- the fact that the lines af, ag, ah already lie in the 

 plane of projection as chosen). We have thus a plane figure, 



consisting- of the fifteen lines ab, ac, gh, and of the 



twenty points abc, abf, fgh; and which is such, that 



on each of the lines there lie four of the points, and through 

 each of the points there pass three of the lines, viz. the 

 points abc, abf, abg, abh lie on the line ab; and the lines be, 

 ca, ab meet in the point abc, and so in the other cases. 



Moreover, from an inspection of the scheme in § 1, we see 

 that the projections of the lines af, bg, ch meet in a point, 

 and the like for each of the six triads of lines; that is, in the 

 plane figure, we have six points 1, 2, 3, 4, 5, 6, each of them 

 the intersection of three lines as shown in the diagram 



1 = af X bgX ch 



2 = agX bh X cf 



3 = ah X bfX eg 



4 = afXbhXcg 



5 = agX bf X ch 

 6=akX bgX cf 



and these six points lie in a conic (the intersection of the 

 quadric cone by the plane of projection). It is clear that the 

 lines af ag. ah) bf, bg, bh; cf, eg, ch are the lines 14, 25, 36; 

 35, 16, 24; 26, 34, 15 respectively. 



Conversely, starting from the points 1, 2, 3, 4, 5, 6 on a 

 conic, and denoting the lines 14, 25, 36; 35, 16, 24; 26, 34, 15 

 (being, it may be noticed, the sides and diagonals of the hex- 

 agon 162435) in the manner just referred to, then it is possible 



to complete the figure of the fifteen lines ab, ac, gh 



and of the twenty points abc, abf, .... fgh, such that each 

 line contains upon it four points, and that through each 

 point there pass three lines, in the manner already mentioned. 



Of the fifteen lines, nine, viz. the lines af, ag, ah; bf, bg, 

 bh; cf, eg, ch are, as has been seen, lines through two of the 



