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Canonical Forms of the Quaternary Cubic associated with arbitrary Quadrics

Published online by Cambridge University Press:  24 October 2008

H. W. Turnbull
H. W. Turnbull
Affiliation:
Trinity College.
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Extract

§ 1. The general quaternary cubic

can be expressed in various interesting canonical forms involving suitably chosen linear forms Xi, Yi, Ai. Thus, referred to the pentahedron. X1X2X3X4X5 of Sylvester the cubic becomes the sum of five cubes

with its Hessian in the form

where the coefficients ai may if necessary be taken as equal to unity and the five linear forms Xi each contain four independent parameters, making a total of twenty parameters which is the number of coefficients aijk in the given cubic C3

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 22 , Issue 2 , May 1924 , pp. 92 - 100
Copyright
Copyright © Cambridge Philosophical Society 1924

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References

* Camb. Math. Journ. 6 (1851), p. 198Google Scholar: Clebsch, , Cretle, 59.Google Scholar

† Reye, , Journ. f. Math. 78 (1874), p. 114.Google Scholar

‡ “On Canonical Forms,” Proc. Lon. Math. Soc. Ser. 2, Vol. 18 (1920).Google Scholar

* Cf. Wakeford, loc. cit.

* Cf. “Sextactic Cones and Tritangent Planes,” Proc. Lon. Math. Soc. Ser. 2, Vol. 21 (1922), p. 375.Google Scholar

* I am indebted to Professor H. F. Baker for this extension of the original theorem, and also for suggesting the verification of G2 (§ 5).

† Richmond, H. W., Quarterly Journal of Mathematics (1902).Google Scholar

* Cf. Baker, , Principles of Geometry, Vol. III, p. 155 (Cambridge, 1923).Google Scholar