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On Five Lines, in Space of Four Dimensions, which lie upon a, Quadric Threefold and the Normal Quartic Curves of which they are Chords

Published online by Cambridge University Press:  24 October 2008

F. Bath
F. Bath
Affiliation:
King's College.
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Extract

The connexion between the conditions for five lines of S4

(i) to lie upon a quadric threefold,

and (ii) to be chords of a normal quartic curve,

leads to an apparent contradiction. This difficulty is explained in the first paragraph below and, subsequently, two investigations are given of which the first uses, mainly, properties of space of three dimensions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 22 , Issue 2 , May 1924 , pp. 189 - 199
Copyright
Copyright © Cambridge Philosophical Society 1924

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References

* Or, the equations of a normal quartic of S 4 contain 25 homogeneously entering coefficients of which 3 may be removed by linear transformation of the parameter— not affecting the curve—so the equations contain 21 effective constants. And, generally, the equations for the normal n–ic contain (n + 1)2 − 4 = n 2 + 2n − 3 effective constants.

† See Prof. Segre, C., Rendiconti del Circolo Matematico di Palermo, Tomo II (1888), p. 45.Google Scholar

* See Prof. Baker's, H. F. paper: “On a proof of the Theorem of a Double Six of lines by projection from Space of Four Dimensions,” Proc. Camb. Phil. Soc. vol. xx (1920), p. 136, § 3, or Prof. Segre's paper cited above.Google Scholar

† To make it quite clear that two conies with a common point form a degenerate normal quartic we may add a construction which obtains such a pair as the residual intersection of three Q 3's through three lines, and their transversal. For, join three points of one conic to three points of the other by three lines: then there are three linearly independent quadric threefolds through

(1) the common point of the two conics,

(2) four further points on each of the conics,

and (3) three further points, one on each of the three lines.

Each of these quadrics meets the conies in five points each and the lines in three points each; therefore contains both conies, the three lines—and their transversal.

‡ Math. Annalen, Bd. v (1872), p. 261Google Scholar, or Gesammelte Mathematische Abhandlungen, Bd. I, p. 111, also p. 153.Google Scholar

* I.e., non-degenerate.

† See Prof. Baker, H. F., Principles of Geometry, vol. III, p. 139, Ex. 19.Google Scholar

* In the sequel ‘quadrie’ will be written for ‘quadric threefold’.

* It is not difficult to show that, if the quartic is not genuine the transversal lines and planes of a, b, c, d, e possess special properties.

† This is stated by MrRichmond, H. W., Proc. Camb. Phil. Soc. vol. x (1899), p. 212.Google Scholar

* See Wakeford, E. K., Proc. Land. Math. Soc. Ser. 2, vol. xv (1916), p. 341Google Scholar. The proof there given uses Miquel's Theorem; also MrWhite's, F P. recent paper in Proc. Camb. Phil. Soc. vol. XXII (1924), p. 11.CrossRefGoogle Scholar

† I.e. there exist triangles whose vertices lie on k and whose sides touch A′ or B′ or C′, etc., as the case may be.

* The equation of a symmetric (2, 2) correspondence is of the form

and the assignment of five pairs of corresponding points determines α: β:…:Φ.

* First discussed by Stephanos, C., Comptes Rendus, t. 93 (1881), pp. 578–580 and 633–636Google Scholar; see also Prof. C. Segre already quoted.

† Wakeford, E. K., “Chords of Twisted Cubics,” Proc. Land. Math. Soc. Ser. 2, vol. xxi, p. 98.Google Scholar

‡ Prof. Segre, C., Memorie di Torino, xxxix (1888), p. 3Google Scholar; Atti di Torino, XXII (1887), p. 791Google Scholar; also MrRichmond, H. W., Quart. Journ. Math, xxxi (1899), p. 125, and xxxiv (1902), p. 117.Google Scholar

* See MrHodgkinson, J., Proc. Land. Math. Soc. Ser. 2, vol. xv (1916), p. 343.Google Scholar