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On the higher singularities of plane algebraic curves

Published online by Cambridge University Press:  24 October 2008

C. A. Scott
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Extract

1. During the last sixty years the principal questions presented by the higher singularities of plane algebraic curves have been completely solved, and definite results obtained. The two most successful lines of research have been by expansions and quadratic transformation. By each method it has been shown that a higher singularity may be looked upon as containing concealed or “latent” multiple points or lines in addition to those immediately recognized; and from each, with the help of small quantities, has been constructed a topological explanation of these latent multiple elements, which are accounted for as situated in the immediate vicinity of the point and line base of the singularity. Further, by each method it has been proved that as regards the numerical relations known as Plvicker's equations a singularity produces the same effect as a definite number of nodes, cusps, bitangents, and stationary tangents.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 23 , Issue 3 , July 1926 , pp. 206 - 232
Copyright
Copyright © Cambridge Philosophical Society 1926

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References

* The vertices of the triangle of reference opposite to the sides x, y, z are distinguished as O 1, O 2, O 3, of which the last is O when non-homogeneous co-ordinates x, y are used. It is convenient to choose the triangle so that O 1, O 2 are not common to F, f; also if F, f have a common tangent at O3, this should not be used as a line of reference.Google Scholar

* This process, which gives at every stage the greatest admissible value for θ, may possibly be the same as that of Stewart, John of Aberdeen (1745) mentioned by de Morgan: “Newton's Method of Coordinated Exponents,” Trans. Camb. Phil. Soc. ix, 1856.Google Scholar

* Cayley, , Quart. Journ. Math. vii. 1865.Google Scholar

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* In Proc. Lond. Math. Soc. vi. 1873–6Google Scholar, Smith, H. J. St. obtains the expansions in line-coordinates, and thus finds l, but as the argument depends on small quantities, it is not admissible hereGoogle Scholar. The same comment applies to the proofs of Halphen, (Mém. prés….à l'Ac. des Sc. de Paris, xxvi. 1874–7)Google Scholar and Stolz, (Math. Ann. viii. 1874–5).Google Scholar

* It is not quite clear how Halphen himself arrived at this, but the context suggests that it was by means of small quantities.Google Scholar

* “Ueber Singularitäten ebener algebraischer Cnrven u.s.w. 1879,“ Math. Annalen, xvi. pp. 348–Google Scholar; Vorles. ü. ebene algebraische Kurven u.s.w. 1925, pp. 263–272.Google Scholar