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Some properties of line congruences

Published online by Cambridge University Press:  24 October 2008

L. Roth
L. Roth
Affiliation:
Clare College.
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Extract

In a previous paper the writer has obtained various formulae for surfaces in higher space by means of Cayley's functional method; and recently the same method has been applied to surfaces on a quadric form in [5] with the object of investigating the properties of line congruences in [3]. The purpose of the present note is two-fold: to indicate a few applications of the formulae mentioned above, and to give direct and (it is believed) novel proofs of some results obtained elsewhere by the functional process. The demonstrations given here differ, save in two cases, from those based on Schumacher's four-dimensional representation which, as it stands, applies only to congruences without double rays and therefore lacks generality. It was James who first noticed that a congruence without singularities has in general a finite number of double rays and is accordingly defined by five, instead of four, independent characters. These characters have been considered in previous work; but for present purposes it is convenient to describe them afresh.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 27 , Issue 2 , April 1931 , pp. 190 - 200
Copyright
Copyright © Cambridge Philosophical Society 1931

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References

* Proc. Camb. Phil. Soc. 25 (1929), 390: this paper is referred to as F.Google Scholar

† Proc. Lond. Math. Soc. (2), 32 (1931), 72: this is referred to as L. The present notation is that of L.Google Scholar

‡ Schumacher, , Math. Annalen, 37 (1890), 100.CrossRefGoogle Scholar

§ James, , Messenger of Math. 55 (1925), 9.Google Scholar

* L. § 3.

* Proc. Camb. Phil. Soc. 26 (1930), 54.Google Scholar

† See Segre, , Encyk. der Math. Wiss. iii C 7, 954Google Scholar

* This application of the Cayley-Zeuthen equations is due to Schumacher, op. cit.

* See F. § 5.

† Segre, , loc. cit.: this proof is due to Fano, Mem. di Torino (2), 51 (1901), 21.Google Scholar

* This gives the number of triple points on the scroll (l).

† See F. § 14, or Proc. Lond. Math. Soc. (2), 32 (1931), 161.Google Scholar

* F., equation (26).

† The method given is substantially that of Schumacher.

‡ See the paper referred to in § 3.

* For a list of the congruences employed here see L. § 4.

† In particular, it may happen that such singularities do not exist at all.

* James, loc. cit., 14.