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252 MATHEMATICS: L. E. DICKSON 
(Fa) z = ax -\- a^w, y = a'^w; 
(La) w= ax, y = (a^ + l)x + a%- 
(Ja) z = (a -\- l)x + (a^ + a)w, y = x + (a"" + a)w. 
Those with the same subscript a are permuted as follows: 
A: (DF)(LH)(EJ), B: (DL)(HE)(FJ), C: (DF)(LJ)(EH). . 
Since the 6.3 imaginary lines and the earher 9 lines give the 27 possible 
lines, we now have all the lines on (7), a fact checked otherwise. 
The automorphs were seen to replace the three lines Ha by all of the 
18 imaginary lines. Hence in studying sets of three coplanar lines 
at least one of which is imaginary, Ha may be taken as one line. The 
following are seen to give all such sets: 
Ha, Fa and x = y, w = z, in w -\- z = a(x + y); 
Ha, Ea, y = z = 0, in z = ay; 
Ha, Da"^, Ja^ + a + i, iu ^ (^^^ + ax) = ay + z; 
Ha, La, X = w = 0, in w = ax; 
Ha, Ja^, Da"^ ^a + i, iu (a^ + 1) + dx) = ay + z. 
Those in the second set alone are concurrent, meeting at (100a). Hence 
there are 18.2/3 = 12 sets of three imaginary non-concurrent coplanar 
Hues and 18.3/2 = 27 sets with two imaginary and one real coplanar 
lines, in 9 of which sets the lines concur (since A, B, C permute the 
pairs HE, LJ, DF). 
Theorem 3. Just 9 of the 27 straight lines on (7) are real. Of the 
45 sets of 3 coplanar lines, the lines in 32 sets are not concurrent and those 
in 13 sets concur. 
5. We consider briefly certain cubic surfaces with singular points. 
One for which 11 is the only singular point is 
x^ + xz^ + xw"^ + y'^w + yw^ + xzw = 0. (8) 
The'' only real points not on it are 1 and 5. There are only ten lines 
on this surface, all being real: 
(a) X = y = z, (h) z = X = y w, (c) x = y = z w, 
(d) y = z = X -\- w, (e) X = y = 0, (f) X = w = 0, (g) X ^ 0, y ^ w, 
(h) w = {),x = z, (i) X = w, y = z, (j) x = w = y + z. 
Of these, / and h alone are in = 0, ^ counting as a double line of 
intersection with (8). The only sets of three coplanar lines are a, 
b, h, in X = z, meeting at 11; c, d, h, in z = x -\- w, meeting at 11; 
e, f, ^, in X = 0, meeting at 3; /, i, j, inw = x, meeting at 8; together 
with the sets of non-concurring lines a, d, i, in y = z; a, c, e, in x = y; 
b, c,j, in y = z + w; b, d, g, in y = X + w. 
