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Plane congruences of the second order in space of four dimensions and fifth incidence theorems

Published online by Cambridge University Press:  24 October 2008

W. G. Welchman
W. G. Welchman
Affiliation:
Sidney Sussex College
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Extract

It is well known that the planes which meet four given lines in a space of four dimensions meet a fifth line, determined by the first four. Also the trisecant planes of a rational quartic curve in [4] which meet a line meet another rational quartic curve. These two theorems are of the same type and may conveniently be called “fifth incidence theorems”.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 28 , Issue 3 , July 1932 , pp. 275 - 284
Copyright
Copyright © Cambridge Philosophical Society 1932

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References

* Segre, , Rendiconti Palermo, 2 (1888), 45.CrossRefGoogle Scholar

† This was proved algebraically by James, , Proc. Camb. Phil. Soc., 21 (1923), 664Google Scholar. Similar theorems were proved by Pieri, , Giornale di mat., 28 (1890), 213.Google Scholar

‡ Segre, , Rendiconti Lincei (5), 30 (1), (1921), 67–71 (71).Google Scholar

§ Segre, loc. cit. f. n. ‡.

* It is possible for a congruence of order two to have ∞2 foci of the second order (Segre, loc. cit.). I shall, however, not consider congruences with singular surfaces.

† Sturm, , Liniengeometrie, 2 (1893), 51Google Scholar; see also Pascal, , Repertorium, II, 2 (1912), 1034Google Scholar, from which the table is taken.

* The result is stated by James (loc. cit.) but without detailed proof.

† The planes of (a) which meet λ2 are the planes of the quadric line cone projecting Γ1 from λ2, and all meet β2.

* The fifth incidence theorems arising from (a) and (b) were given by James and Pieri respectively (loc. cit.), but it does not seem to have been noticed that the residual curve in each theorem is the same.

† See § 7.

* This is one of James' results. Loc. cit.

† Cp. § 6·3.

‡ Cp. Pieri, loc. cit.

§ Cp. James, loc. cit.

∥ Cp. Pieri, loc. cit.

* Cp. James and Pieri, loc. cit.

† Welchman, , Proc. Camb. Phil. Soc., 28 (1932), 206.CrossRefGoogle Scholar

* See Pascal, loc. cit., p. 1036.

† For the analogous property of a net of quadric surfaces in [3] see Hesse, , Journal für Math., 49 (1855), 279–332.Google Scholar