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Simplexes and other configurations upon a rational normal curve

Published online by Cambridge University Press:  24 October 2008

F. P. White
F. P. White
Affiliation:
St John's College
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Extract

The theorem that if two triangles be inscribed in a conic their six sides touch another conic is, of course, to be found in all the text-books; it is apparently due in the first place to Brianchon. The further remark, that if three triangles be inscribed in a conic the three conics obtained from them in pairs have a common tangent, is to be found in Taylor's Ancient and Modern Geometry of Conics; it was made independently by Wakeford.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 23 , Issue 8 , October 1927 , pp. 882 - 889
Copyright
Copyright © Cambridge Philosophical Society 1927

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References

* Brianchon, , Mémoire sur les lignes du second ordre, Paris, 1817, p. 35Google Scholar (reference taken from Encykl. Math. Wiss., III, C. 1, p. 35, fn. 100Google Scholar).

† Cambridge, 1881, p. 360. Taylor refers to Picquet, H., Étude géométrique des, systèmes ponctuels, Paris, 1872, but this author only gives the converse.Google Scholar

‡ Von Staudt, , Beitrage zur Geometrie der Lage, 1860, p. 378.Google Scholar

§ Announced in Math. Ann., 15, 1879, p. 14;Google Scholar proof ibid., 20, 1882, p. 135.

‖ Cremona, , Rendiconti Lombardo (2), 12, 1879, pp. 347–52;Google ScholarOpera, t. III, p. 441.Google Scholar

Pasch, , Journal für Math., 89, 1880, p. 256.Google Scholar

** Weyr, , Bulletin de l'acad. roy. de Belgique (3), 3, pp. 472–85.Google Scholar

†† See list in the Royal Society Catalogue of Scientific Papers.

* Meyer, , Apolarität und rationale Curven, 1883, p. 387.Google Scholar

† Encykl. Math. Wiss., III, C 7, p. 896.Google Scholar

‡ See reference in Segre, loc. cit.Google Scholar

§ Kubota, , Science Reports, Tohoku Imperial University, 15, 1926, pp. 39–44;Google ScholarMath. Zeits., 26, 1927, pp. 450–6.Google Scholar

‖ Veronese, , Math. Ann., 19, 1881, pp. 161–234, especially pp. 219–20.CrossRefGoogle Scholar

The proof of the theorem on the twelve faces of three tetrads on a space cubic curve from four dimensions is indicated in Baker's, Principles of Geometry, vol. IV, 1925, p. 147.Google Scholar

* Schur, , Math. Ann., 18, 1881, pp. 1–32.CrossRefGoogle Scholar

* Schur, , loc. cit.Google Scholar

† Jessop, , Quartic Surfaces, Cambridge, 1916, Chap. IX.Google Scholar

‡ The points P in S through which pass trisecant planes of the sextic curve describe a symmetroid, the 10 nodes of which are the points of intersection of S with the variety of chords of the sextic. For the four hexads we get a plane in the six-dimensional space and on it a quartic curve, through the points of which pass trisecant planes; with five hexads we get a straight line and four points of it. The symmetroid, the quartic curve and the four points are the intersections of 5, the plane and the line respectively with the variety of trisecant planes of the sextic, which is of dimension 5 and order 4.

§ Segre, , loc. cit., p. 896, fn. 375.Google Scholar

* The dual is a surface of order n − 1 arising as the locus of intersections of corresponding planes of two projectively related systems of primes through two lines.

* Cf. Segre, , loc. cit. He does not remark that the quadric is degenerate if n > 3. For n = 3, 2 the cone becomes an ordinary quadric and a conic, respectively.+3.+For+n+=+3,+2+the+cone+becomes+an+ordinary+quadric+and+a+conic,+respectively.>Google Scholar

† The result, included in this for n = 3, that, for four tetrads on a cubic curve, the four quadrics touching the faces of threes have a common generator, was remarked by Mr J. H. Grace.