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

22

THE DOUBLE SIX CONFIGURATION

where we set
c1f2 + a2fi + blg2 4 b2gl + c1 2 + C2h = 12, etc.*
This is also the condition that these lines may lie in a cubic
surface t.
The agreement between this equation of condition and that of
Sylvester (A = 0 of ~ 7) inheres in the fact that Cayley's determinant
of the fifth order above written is the square root of Sylvester's A^.
10. Enumeration of the Double Sixes of a Cubic
Surface.
It has been shown (~ 3) that the configuration of twenty-seven
lines contains two hundred and sixteen pairs of non-intersecting
straight lines. Each pair determines a double six; but each double
six contains six pairs of such mated lines. Hence the number of
216
double sixes is 2 = 36.
6
Let us next proceed to form a table of.the double sixes. The
original double six
1, 2, 3, 4, 5, 6
1', 2', 3', 4', 5', 6'
is the primitive. There is but one of this type.
Consider next the type
1, 1', 23, 24, 25, 26,
2, 2', 13, 14, 15, 16.
The number of this type is clearly 15 (= C2).
Of the type given by
1, 2, 3, 56, 46, 45,
23, 13, 12, 4', 5', 6',
there are 20 (= 6C3), since keeping 1, 2, 3 fixed, the remaining numerals
are uniquely determined~. This concludes the enumeration, since
1 + 15 +20 = 36.
Below is the table of all the double sixes.
* Cayley, Coll. Mlath. Papers, Vol. vii. (1867), pp. 66-98.
t Cayley, Coll. Math. Papers, Vol. VII. (1870), p. 178.
+ Sylvester, Comptes Rendus, Vol. LII. (1861), p. 816.
~ In his paper "A Memoir on Cubic Surfaces," Philos. Trans. Royal Soc. Vol.
CLIX. (1869), pp. 231-326, Cayley erroneously states that there are twenty of the type
1, 2, 3, 56, 46, 45,
23, 13, 12, 4, 5, 6.