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The complete system of the binary (3, 1) form

Published online by Cambridge University Press:  24 October 2008

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

1. The only double binary forms for which the complete systems of invariants and covariants are known appear to be the (1, 1), (2, 1) and (2, 2) forms, for which the complete systems were determined sixty years ago by Peano. In the present paper we determine the complete system of the binary (3, 1) form and establish its irreducibility. The system proves to contain twenty forms, and is thus scarcely more complicated than that of the (2, 2) form which includes eighteen concomitants, and much simpler than that of the (2, 1, 1) form derived by Gilham. It is hoped in a subsequent note to interpret the system geometrically.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 42 , Issue 3 , October 1946 , pp. 196 - 205
Copyright
Copyright © Cambridge Philosophical Society 1946

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References

* Peano, , Giorn. matematico, 20 (1882), 79–100.Google Scholar For further references, including simultaneous systems, see Turnbull, , The Theory of Determinants, Matrices, and Invariants (Blackie, 1928), p. 243.Google Scholar

† Gilham, , Proc. London Math. Soc. (2), 32 (1931), 259–72CrossRefGoogle Scholar and 38 (1935), 271–2.

* See Grace, and Young, , Algebra of Invariants (Cambridge, 1903),Google Scholar Chapter VI. The methods and theorems established for simple binary forms extend immediately to multiple binary forms. See, for example, Turnbull, , Proc. Edinburgh Math. Soc. 40 (1923), 116–27.Google Scholar

* We shall often use symbols c, c 1, c 2, … in this sense. Such a symbol appearing in different contexts need not necessarily denote the same constant.