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Published online by Cambridge University Press: 20 January 2009
Gordan's Theorem, that the complete system of irreducible concomitants of a given form is finite, has been extended by Hilbert to cover wide ranges of systems of variables Gordan and Study have dealt shortly with the problem for double binary forms, approaching the subject through the theory of binary types. The following pages give a proof after the manner of Gordan's proof for ordinary binary forms, which has the advantage of providing a practical method for constructing the complete system. As illustrations, the cases of the (1, 2), (2, 2), and (3, 3) forms are considered.
* Math. Ann., Bd. 30 and 36. Cf. Maurer, , “Ueber die Endlichkeit der Invarianten Systeme.” München. Sitzungaberichte der Math. Bd. 29, 1899.Google Scholar
† Cf. Math. Ann., Bd. 33, pp. 387–389 Google Scholar; also Sitz. berich. der Phys.-med. Soc. Erlangen, (1888), p. 35.Google Scholar
‡ Ibid., p. 31.
Cf. Grace, and Young, , Algebra of Invariants, pp. 326–338.Google Scholar
* I=0 unless n, n′, ½(n + n′) are all even.
* Cf. Peano, , Giornale di Math., Battaglini, Vol. xx. Google Scholar, who reaches this result by elementary methods. The treatment of the (2, 2) form is very thorough.
* Proc. Roy. Soc. Edinburgh. Vol. xliii., p. 50.Google Scholar
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