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On the number of planes in hyperspace which satisfy a certain set of conditions

Published online by Cambridge University Press:  24 October 2008

J. A. Todd
J. A. Todd
Affiliation:
Trinity College
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Extract

The problem here considered is a particular case of the more general problem of finding the number of [k]s, lying in a space of dimension n(n > k), which satisfy a certain number of conditions of the following type: viz. to meet a given [a0] in a point, a given, [a1] containing the [a0], in a line, etc., finally to lie in a given [ak] containing [ak−1]. The number of such conditions is such that there are just a finite number of [k]s which satisfy them all. The method which we employ is the one introduced by Schubert and commonly known as the “degeneration method” it is explained very briefly below.

Type
Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 25 , Issue 1 , January 1929 , pp. 42 - 47
Copyright
Copyright © Cambridge Philosophical Society 1929

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References

* We denote as usual by [k] a. flat space of k dimensions.

† Schubert, , Kalkül der Abzählende Geometrie, Leipzig, 1879.Google Scholar

‡ Bertini, , Introduzione alla geometria proiettiva degli iperspazi (1923), p. 39.Google Scholar

§ Bertini, , loc. cit. p. 38.Google Scholar

* Bertini, , loc. cit. p. 254.Google Scholar

† Bertini, , loc. cit. Chap. 4.Google Scholar

* Encyk. der Math. Wiss. III C 7, p. 817, footnote 157.Google Scholar

† Mem. Acc. Torino (2), 52 (1902), p. 171.Google Scholar

‡ I am indebted to Mr W. R. Andress for this remark.