Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-29T22:17:32.707Z Has data issue: false hasContentIssue false
  • English
  • Français

A figure in space of seven dimensions, and its sections

Published online by Cambridge University Press:  24 October 2008

W. L. Edge
W. L. Edge
Affiliation:
Trinity College
Get access Rights & Permissions [Opens in a new window]

Extract

Several papers have recently been published which include proofs of the following:

Theorem 1. Those trisecant planes of a rational normal quartic curve Cu which meet a second rational normal quartic curve Cv having six points in common with Cu also meet a third rational normal quartic curve Cw through these same six points. The three suffixes may be permuted in any way, the three curves forming a symmetrical set.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 30 , Issue 1 , January 1934 , pp. 19 - 26
Copyright
Copyright © Cambridge Philosophical Society 1934

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

* Welchman, W. G., Proc. Camb. Phil. Soc. 28 (1932), 275CrossRefGoogle Scholar; H. G. Telling, Ibid. 403; D. W. Babbage, Ibid. 421; Semple, J. G., Journal London Math. Soc. 7 (1932), 266.CrossRefGoogle Scholar

† James, C. F. G., Proc. Camb. Phil. Soc. 21 (1923), 673.Google Scholar

* Cf. Edge, , Proc. London Math. Soc. (21), 33 (1931), 53.Google Scholar

* The locus is rational and is represented on [3] by means of cubic surfaces through three skew lines, and in other ways.

* A referee points out that these three systems of [5]'s are dual to the three systems of lines on . In this connection see also James, loc. cit. 674–675. Just as a trisecant plane of determines, by means of the three directrices through its intersections with , a secant [5], so a [4] which meets three generating solids of each in a line contains a directrix of , namely the transversal of the three lines. Viewed from this aspect Theorem 2 is the dual of the statement that any [4] which contains a directrix σ of lying on meets some generating solid of in a line and some generating solid of in a line. The truth of this statement is obvious; both the lines in question are the line σ itself, which is the dual of the space σ6 of Theorem 2.

* This configuration of three lines and three twisted cubics occurs in Welchman's paper: Proc. Camb. Phil. Soc. 28 (1932), 416–420 (418).Google Scholar

* Cf. Telling, loc. cit. 415.

† Segre, , Atti Torino, 19 (1884), 355.Google Scholar

‡ Cf. Baker, , Principles of Geometry, 4 (Cambridge, 1925)Google Scholar, chapter 5; Semple, loc. cit. 270.

* Segre, , Math. Annalen, 27 (1886), 296–314 (302).CrossRefGoogle Scholar