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Enumerative Formulae for Surfaces

Published online by Cambridge University Press:  24 October 2008

L. Roth
L. Roth
Affiliation:
Clare College
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Extract

The formulae considered in this paper deal with the multiple tangent lines and planes to surfaces in three dimensions; the functional method employed is that of a previous paper, in which a more restricted discussion was given for surfaces in higher space.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 26 , Issue 1 , January 1930 , pp. 43 - 71
Copyright
Copyright © Cambridge Philosophical Society 1930

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References

* Proc. Camb. Phil. Soc. 25 (1929), 390–406; this is referred to as F.CrossRefGoogle Scholar

† See Salmon-Rogers, , Analytic Geometry of Three Dimensions, II (1915); this is referred to later as Salmon.Google Scholar

‡ Schubert, , Math Ann. 10 (1876), 98;Google Scholaribid. 11 (1877), 347; 12 (1877), 180. See also Kalkül der Abzählenden Geometrie (1879).

§ James, , Trans. Camb. Phil. Soc. 23 (1925), 201; this is referred to as J.Google Scholar

∥ Cayley, , Coll. Papers, vi, 359;Google Scholar Schubert, op. cit.; Basset, Geometry of Surfaces, Ch. ix (referred to as Basset).

Cayley, , Coll. Papers, vi, 582, xi, 225;Google ScholarZeuthen, , Math. Ann. 10 (1876), 446 (this paper is referred to as Zeuthen).CrossRefGoogle Scholar

* Basset, , Quart. Journal, 42 (1911), 225.Google Scholar

† Cayley, , Coll. Papers, vi, 347.Google Scholar

* See Zeuthen, p. 450; Salmon, p. 314.

† See F. §§ 1, 13; the surface F 7, which is described in § 1, has one improper node, not two, as stated.

* Basset, p. 31.

′ It does not vanish for F 4′ on account of the nodal lines through the triple point.

* Salmon, p. 287; the coefficient −12 should be +12.

† This method, which is used by James, seems to have originated with Voss, , Math. Annalen 9 (1876), 483.CrossRefGoogle Scholar

‡ In this argument, all ε-considerations have been omitted in order to save unnecessary complexity.

* Basset, p. 40.

* For a scroll the multiplicity is only double and the curve counts eight times.

* Schaake, , Proc. Roy. Soc. Amsterdam, 27 (1924), 362–370.Google Scholar

* Its tangent planes along Γ are the tangent planes to F.

† The correspondence given by James (J. p. 223) is misprinted; the coefficient of Z should be 2.

* Salmon, p. 162.

* Salmon, p. 302.

† Basset, p. 39.

‡ Zeuthen, p. 469.

* Basset, p. 196.

* The dual method is due to James (J. p. 216); the coefficient of n4 should be unity in his result.

† Basset, p. 196.

* Basset, p. 283; the coefficient − 11 should be − 12.

* Basset, p. 273.