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XXI.—Sylvester's Unravelment of a Ternary Quartic

Published online by Cambridge University Press:  14 February 2012

W. L. Edge
W. L. Edge
Affiliation:
Mathematical Institute, University of Edinburgh
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Summary

The paper makes use, for the study of a ternary quartic, of a five-dimensional configuration consisting of a Veronese surface and a quadric outpolar to it, and uses the notation and results of a preceding paper to which reference is made at the outset. In § 1 certain identities are given which are consequences of the form of the matrix of a quadric outpolar to a Veronese surface, and the geometrical theorems equivalent to these identities are stated. In § 2 it is explained how covariants and contravariants of a ternary quartic are represented by curves in the fivedimensional configuration. It is, indeed, not until this technique is used that some of the work of Clebsch, Ciani, Coble, and others is properly appreciated; §§ 3—6 are concerned to emphasise this. But, as is pointed out in § 7–9, it is to Sylvester that these matters must properly be referred; for he has, by his process of unravelment, anticipated practically everything of moment in the ideas of his successors. The word unravelment is used by him on p. 322 of Vol. I of his Mathematical Papers, the process having appeared on p. 294.

In opening the second part of the paper with § 10 it is pointed out that the configuration should be used not merely to illuminate the work of previous writers, but also to discover new results. It is not the purpose here to exploit this at length, but it is seen how a covariant conic inevitably appears; its equation is obtained and, in §11, its covariance directly established. Other covariant conies are alluded to in § 12. And it is found, in § 13, that here too reference must be made to Sylvester.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 61 , Issue 3 , 1943 , pp. 247 - 259
Copyright
Copyright © Royal Society of Edinburgh 1943

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References

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