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.

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Title
The twenty-seven lines upon the cubic surface ... by Archibald Henderson.
Author
Henderson, Archibald, 1877-1963.
Canvas
Page 12
Publication
Chicago,
1915.
Subject terms
Surfaces, Cubic

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"The twenty-seven lines upon the cubic surface ... by Archibald Henderson." In the digital collection University of Michigan Historical Math Collection. https://name.umdl.umich.edu/abr1416.0001.001. University of Michigan Library Digital Collections. Accessed May 29, 2025.
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