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Singularities of Curves and Surfaces

Nature volume 85, page 336 (12 January 1911) | Download Citation

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Abstract

THERE is a distinction between multiple points and what, for want of a better word, I have called singular points. The curve αup + up+1=0 has at A a multiple point of order p, but not a singular point. The latter points are defined in § 169 of my “Geometry of Surfaces”, reviewed in NATURE of December 22, 1910 (p. 231), and the definition may be illustrated as follows. Let multiple points of orders p. q, r ..., where p is not less than q, r ..., move up to coincidence along a continuous curve; then the compound singularity thereby formed is a singular point of order p. The curve of lowest degree, which can possess a singular point of given order, depends on the way in which the singularity is formed. Thus if four nodes move up to coincidence along a conic, the resulting singular point is of the second order; but a quintic is the curve of lowest degree which can possess such a singularity. Also, if three nodes move up to coincidence along a straight line, the singular point is still of the second order, but no curve of lower degree than a sextic can possess such a point.

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  1. December 23, 1910.

    • A. B. BASSET

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DOI

https://doi.org/10.1038/085336a0

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