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Combinants of a pencil of quadric surfaces. I

Published online by Cambridge University Press:  24 October 2008

J. A. Todd
J. A. Todd
Affiliation:
Trinity CollegeCambridge
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Extract

The classical account of the invariants and covariants of a pair of quadric surfaces is due to Salmon. The actual determination, in explicit form, of the complete system of concomitants of two quaternary quadratic forms is much later in date, and is due to Turnbull. This system, which includes mixed concomitants of various kinds, is complicated. As originally determined, it comprised 125 forms, three of which were later shown to be reducible. In the accounts of both these authors, however, the forms are considered as belonging to a particular pair of quadric surfaces, and the problem of determining these covariant forms which are invariant, not merely under change of coordinate system but also under change of base in the pencil denned by the two quadrics, is not alluded to. Such forms, which are of obvious geometrical interest, are called combinants. It is rather surprising to find that very little seems to be known about the combinants attaching to a pencil of quadric surfaces; the Encyklopädie scarcely mentions them, and I have been unable to trace any references in the literature except to the two invariants whose vanishing expresses the condition that the parameters of the four cones in the pencil form an equianharmonic or a harmonic set.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 43 , Issue 4 , October 1947 , pp. 475 - 487
Copyright
Copyright © Cambridge Philosophical Society 1947

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References

* Salmon, , Analytical Geometry of three dimensions, 1 (6th ed. 1914), Chap. IX.Google Scholar

† Turnbull, , Proc. London Math. Soc. (2), 18 (1919), 69.Google Scholar

‡ Williamson, J.London Math. Soc. 4 (1930), 182. I understand that Turnbull has recently found that five further forms are reducible.Google Scholar

* Todd, , Proc. London Math. Soc. (in the Press).Google Scholar

* I am indebted to a referee for suggesting this proof. The argument clearly applies to any pencil of forms (not merely to quadratics).Google Scholar

† Grace, and Young, , Algebra of Invariants (Cambridge, 1903), p. 55.Google Scholar

* Cf. Elliott, , Algebra of Quantics (2nd ed.Oxford, 1913), p. 125.Google Scholar

* See Segre, B., Proc. Cambridge Phil. Soc. 40 (1944), 121.CrossRefGoogle Scholar

* Salmon, , loc. cit. ante, 241.Google Scholar