250 
MATHEMATICS: L. E. DICKSON 
The twelve imaginary lines in which b and c are integers, while 
+ a + 1 =0, are seen to be permuted transitively by (zw), tt, jS. 
We now have 15 + 12 lines on (1). As in the algebraic theory, there 
are only 27 lines on a cubic surface without singular points.^ Hence 
the 27 lines are permuted transitively by / and the real automorphs. 
To prove Theorem 2 below, it therefore suffices to take A as one of 
the three coplanar lines. The only planes through A are z = 0 and 
X = kz. The former cuts the surface in the lines A, z = y = 0, and 
z = 0, X = y, which concur at 4. The latter cuts the surface in the 
lines A and 
X = kz, ky(y + kz) + wiz + w) =0. 
This quadratic function has the factor w ay + I3z if and only if 
13" = ^, k = a^, a = k^. Hence A meets just 10 lines on the surface, 
the above two and 
Lj,^: X = kz,w = k'^y + ^z (i3 = 0, 1; k^ = k). 
The only coplanar sets of three lines, one of which is A , are the above 
set of three real and the four sets A, Lko, Lki, which meet at (010^^)^ 
In all there are 27.5/3 = 45 sets of coplanar lines. 
Theorem 2. The 27 straight lines on (1) form 45 sets of three 
coplanar lines. The three of each set concur and their plane is the tangent 
to the surface at the point of concurrence. 
3. There are several types of real cubic surfaces the numbers of 
whose real points and real lines differ from those of (1), but are de- 
rivable from (1) by imaginary linear transformation and hence have 
45 sets of three coplanar concurrent straight Knes. 
If in (1) we replace z by ez e^w and w hy e^z -\- where 
e^ + e -\- \ =0, and hence replace z -\- w by e^z -\- ew, we evidently 
get 
xy(x -\- y) = z^ -\- zw^ w^. (2) 
For integral values of the variables, the left number is zero modulo 2, 
whence z ^ w = 0. Thus 1, 2, 5 are the only real points on (2); it 
contains no real Hne. 
If in (1) we make also the like replacement of x, y, we get 
x^ 4- xy^ + y^ = z^ + zw^ + w^. (3) 
It contains just nine real points and only three real lines : z = x, y = w; 
z = y = x + w;w = x = y-{-z, no two of which intersect. 
If in (1) we replace^ s by s + bw and why z-\- b^w, where b^ -{-b-\- 1=0, 
we get 
xy(x + = w{z'^ + zw -i- w^). (4) 
