MATHEMATICS: L. E. DICKSON 
The only real points on it are evidently the seven in the plane w 0. 
Hence the only real lines on it are the three having w = 0. 
If in (2) we replace x by x -\- by and y by x -\- b'^y, where 
&2 + 6 + 1 = 0, we get 
y(x^ -\- xy -\- y^) = 2^ + zw"^ + "^^^ (5) 
Replacing y by y z w, z by z + y, w by w -\- y, we get a surface 
whose real points are those on the cone yz ^ yw -\- zw = 0 and hence 
are 1, 2, 3, 4, 5, 6, 7. Thus the real points are those on the lines 
y = z = 0, y = w = 0, z = w = 0, meeting at 1. 
If in (1), written in Capitals, we set 
X = X -{-y, Y = bx + b'y {a^ + l)w, Z = ax + (a -j- l)y + z, 
W = w, _j_ ^, -|_ 1 = 0, + a + 1 = 0, 
we get the real surface^ 
^3 _^ y _j_ (^2 4_ ^2)^ + {y + z)w^ = 0, (6) 
which has only eleven real points and only five real lines 
w = 0, y = x; z = X -\- tw, y = x; z = tw, y = x + w = 0, 1). 
4. A surface without singular points and not having every set of three 
coplanar Hnes concurrent, as was the case with all the preceding sur- 
faces, is 
xy(x -\- y) = w{w -\- z){y z). (7) 
It contains thirteen real points, 9 and 12 alone being not on it. We 
obtain a straight line on (7) by equating to zero one factor of each 
member. The resulting nine Knes will be shown later to be the only 
real Hnes on (7). The three in x = 0 are not concurrent, likewise the 
three in x = y, while the three in y = 0 meet at 1. We obtain a re- 
distribution of these nine lines by beginning with a factor of the second 
member of (7): the three Knes in w = 0 meet at 3, the three in w = z 
meet at 10, the three in y = 2 meet at 4. There is no plane other than 
these six which contains three coplaner lines from this set of nine real 
lines. 
Evident automorphs*^ of (7) are 
(A) = z -\- y -\- w; (B) = w -\- z; (C) = x y. 
Evident imaginary lines on (7) are 
(Ha) w = ax, z = ay (a^ + + 1 = O). 
Applying the real automorphs to Ha, we get the new lines 
(Da) z = ax -\- w, y = aHv; 
(Ea) z = ax -\- w, y = X + (a -\- a'^)w; 
