Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-24T17:01:10.954Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiple tangents to forms in four dimensions

Published online by Cambridge University Press:  24 October 2008

L. Roth
L. Roth
Affiliation:
Clare College
Get access Rights & Permissions [Opens in a new window]

Extract

Formulae for multiple tangents to the general surface in ordinary space were obtained at different times by various writers, but the discussion remained incomplete until the advent of Schubert's enumerative method, which solves the whole problem by a purely mechanical process. Schubert later extended the method to general forms in [n], in the restricted case where the multiple tangents have only a single contact. In the present paper the results for three dimensions have been used to build up the formulae for forms in [4], by means of the correspondence theorem on which much of Schubert's work is based. It would no doubt be possible to evolve a complete set of incidence formulae for four dimensions and then to proceed as in Schubert's discussion of surfaces; but the present method is preferable for two reasons. In the first place, all the results have been obtained in a very simple manner; and secondly, a large number of minor results have been found in the process.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 26 , Issue 3 , July 1930 , pp. 285 - 304
Copyright
Copyright © Cambridge Philosophical Society 1930

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

* Schubert, , Math. Annalen, 11 (1877), 347.CrossRefGoogle Scholar

† Schubert, , Math. Annalen, 26 (1886), 52. (This is referred to as Schubert)CrossRefGoogle Scholar

‡ See e.g. Zeuthen, , Lehrbuch der Abzäihlenden Methoden (Leipzig, 1914), 389.Google Scholar

§ See Salmon-Rogers, , Analytic Geometry of Three Dimensions, II, 289.Google Scholar

* There are, however, exceptions; see § 5·21.

* The first suffix is used, for brevity, to denote the number of simple contacts.

* Schubert, 72.

* Schubert, 72.

† It is noteworthy that this result and (44) both vanish for F4.

* Schubert, 71.