The twenty-seven lines upon the cubic surface ... by Archibald Henderson.

THE DOUBLE SIX CONFIGURATION

15

Suppose that A, B, C, D, E are the other lines intersecting (b, c, d, e),
(c, d, e, a), (d, e, a, b), (e, a, b, c), and (a, b, c, d) respectively. Then
A, B, C, D, E zill all be met by one other straight line x.
The double six in this case is written
a, b, c, d, e, x
A B, C, D, E, X
Schlafli then proposes the question: "Is there, for this elementary
theorem, a demonstration more simple than the one derived from the
theory of cubic forms? "
Sylvester* states that the theorem admits of very simple geometrical proof; but he did not supply the proof. Salmont has given a
method for constructing a double six, by pure geometry; but it is not
a proof of the theorem, independent of the cubic surface.
In 1868, Cayley gave a proof of the theorem from purely static
considerations, making use of theorems on six lines in involution.
It has recently been remarked, by Mr G. T. Bennett, that this is
erroneous~. Again in 1870, Cayleyll verified the theorem, using this
time the six co-ordinates of a line. In 1903, Kasner~T also gave a
proof using the six co-ordinates of a line. More recently (January 13,
1910), Baker has given a direct algebraic proof of the theorem independently of the cubic surface, so formulated as to show that the
theorem belongs to three dimensions only*".
In 1881, Schurtt originally gave a geometrical proof of the double
six theorem, basing his proof on a poristic property of the plane cubic
curve. Recently (November 21, 1910), Baker It has given a geometrical
proof of the double six theorem independently of the cubic surface,
thus demonstrating the fundamentally projective character of the
configuration.
* "Note sur les 27 droites d'une surface du 3e degre," Comptes Rendus,
Vol. LII. (1861), pp. 977-980.
t Geometry of Three Dimensions, 4th edition, p. 500.
+ "A 'Smith's Prize' Paper, Solutions," Coll. Math. Papers, Vol. vmII. (1868),
pp. 430-431.
~ "The Double Six," Proc. London Math. Soc. Ser. 2, Vol. Ix. (1911),
p. 351.
11 "On the Double Sixers of a Cubic Surface," Quart. Journ. Vol. x. (1870),
pp. 58-71.
~T American Journ. of Math. Vol. xxv. No. 2, pp. 107-122.
i* Proc. London Math. Soc. Ser. 2, Vol. Ix. Parts ii. and III. pp. 145-199.
ft+ Math. Ann. Bd. xvIIi. pp. 10, 11.
++ Proc. Boy. Soc. A, Vol. LXXXIV. pp. 597-602.