MATHEMATICS: L. E. DICKSON 
253 
Theorem 4. There are only ten lines on (8) and all are real. Three 
coplanar lines concur if and only if one of them belongs to the singular 
pair /, h. 
In (8) we replace x hy y, z hy z -\- y, y hy x -\- y -\- ew, where 
^2 + e + 1 = 0; we obtain the real surface 
having only ten straight lines of which only two are real. The five real 
points are those for which yz = w = 0, viz., 1, 2, 3, 5, 6. 
The only type other than (9) having 1, 2, 3, 5, 6 as its only real points 
and having a singular point is + x'^y + xy"^ + yz^ + y^w = 0, con- 
taining only three straight lines: y = w = 0; y = w, z = x -\- bw, 
b^ b 1 =0; these meet at the only singular point 6. 
Another type with only five real points is xw^ = y^ + yz^ + 2', the 
only lines on which are the six in the planes x = 0, = 0. 
1 And on only the surfaces equivalent to (1) or to xyix -\- y) = ^ under real linear 
transformation. 
2 They generate all of the 15.6.4.2 real linear automorphs. For, one leaving point 2 fixed 
must permute the remaining real points 3, 4, 8, 9, 10, 14 in the tangent plane = 0 at 2. 
These six are permuted transitively by (zw) and 7, both of which leave 2 fixed. An auto- 
morph which leaves 2 and 3 fixed, and hence the point 8 coUinear with them, must per- 
mute 4, 9, 10, 14. These are permuted transitively by /3 and •y^izw), which leave 2 and 3 
fixed. An automorph leaving 2, 3 and 4 fixed is the identity ox: = y -\- x, which is the 
transform of ^ by (zw)^. 
3 We readily verify that A^B^C and Eab, Dahc, with h and c integers and a* = a, give 
all of the 27 lines on (1). We have only to consider first the lines whose two equations are 
solvable for y,z in terms of x^w, and second the lines one of whose equations is a: = 0 or 
w = ax. 
* If we make also the like replacement on x,y, we change the left member of (4) into 
y{x'^ -\- xy y2). The new surface evidently has its seven real points in y — w. Replacing 
why w -\- y, xhy X -\r z and then z by z + y, we get (4). 
^ Its 16 real linear automorphs are generated by the four: z^ = z + w; = y + w, y = 
X + w; x^ = y, y^ = X, = X -\- y -\- z; and y^ = y + 7e', z^ = y + z. 
^They generate all the twelve real linear automorphs of (7). For, such a transforma- 
tion must leave fixed the pair of points 9, 12 and hence the tangents x = y, a; = 0 at those 
points. These points and planes are interchanged by C. If each is fixed, the transforma- 
tion is 
Since the lines in a; = 0 are permuted, the case / = 1, g = 0, is excluded. For/ *= g = 0? 
we get the identity and A. For / = 0, g = 1, we gtt B, BA. For / = g = 1, we get 
AB, BAB. 
' As shown in the earlier paper, any cubic surface with just 13 real points and having no 
linear factor is equivalent to (7) or (8). 
(9) 
x^ = X, y^ = y, z^ = by -\- cz + bw, = fy gz (/+ l)w. 
