MATHEMATICS: L. E. DICKSON 
249 
Theorem 1. This surface with exactly 15 real points contains exactly 
15 real straight lines. There are exactly 15 sets of three coplanar real 
lines on the surface, the three of each set being concurrent, while their plane 
is the tangent to the surface at the point of concurrence. 
The surface has the automorphs^ (i.e., transformations carrying the 
surface into itself) 
{zw) ; TT = {xz) {yw) ; 13: z^ = z -\- w; 
y: x^ = X, y^ = y w, z^ = z -{- x, w^ = x z w. 
The fifteen real points of space 
1 = (1000) 5 = (1100) 9 = (0101) 13 = (1011) 
2 = (0100) 6-= (1010) 10 = (0011) 14 = (0111) 
3 = (0010) 7 = (1001) 11 = (1110) 15 = (nil) 
4 = (0001) 8 = (0110) 12 = (1101) 
are permuted transitively by these automorphs, since 
(zw): (3 4) (6 7) (8 9) (11 12), 
P: (4 10) (7 13) (9 14) (12 15), 
t: (1 3) (2 4) (5 10) (7 8) (11 13) (12 14), 
7: (1 13 12 6) (3 10 8 14) (4 9) (5 15 7 11). 
The tangents to (1) at the fifteen real points give all of the fifteen 
real planes in space, since the tangent at (XVZW) is 
Y^x + X^y + W^z + Z'^w = 0. 
Hence these planes are permuted transitively by the real linear auto- 
morphs of the surface. The plane x = 0, tangent at 2, contains the 
concurrent lines 
A: X = z = 0, B: X = w = 0, C: x = 0, z = w, 
evidently on the surface. Hence each of the fifteen real planes con- 
tains three (concurrent) fines lying on the surface (1). The only real 
planes through line A are evidently x = 0, z = 0, and x = z. Hence 
the number of real lines on (1) is 15.3/3. We have therefore proved 
Theorem 1. 
Since (zw) and (3 replace ^ by and C and since the fifteen real 
planes are permuted transitively by the automorphs, the fifteen real 
lines on the surface are permuted transitively by the real linear auto- 
morphs. 
We next make use of the imaginary automorph / which leaves z 
and w unaltered, but multipHes x and y by t, where t^-{-t -{- 1 = 0. It 
replaces the line w = x, y = z on (1) by (w = tx, z = ty), where 
Eab : w = ax, y = bx + a%' Dabc : z = ax -\- bw, y = -{- a^. 
