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Published online by Cambridge University Press: 15 September 2014
In the following pages we propose to clear up certain difficulties which have arisen in the theory of invariants belonging to two quadratic forms homogeneous in n variables x1, x2, …, xn. We denote such a set by a single symbol x, and take it as fundamental that any non-zero set x represents a point in space of n – 1 dimensions. We deal with any number of cogredient sets, x, y, z, x1, x2, etc., each representing a point, of which n at most are linearly independent in this space Sn–1. A prime u is a space Sn–2, lying wholly in the Sn–1, and therefore determined by n–1 linearly independent points.
page 9 note * Turnbull, , Trans. Camb. Phil. Soc., 21 (1909), 197–240.Google Scholar
page 9 note † Turnbull, and Williamson, , “The Minimum System of two Quadratic Forms” (Proc. Roy. Soc., Edin., 45 (1925), 149–165)CrossRefGoogle Scholar
page 11 note * Turnbull, H.W., The Theory of Determinants. Matrices and Invariants (1928), p. 203.Google Scholar (Henceforward this will be called ‘Invariants.’)
page 11 note † Cf. Invariants, p. 81.
page 12 note * Invariants, p. 45, I.
page 14 note * In future we shall usually omit this numerical non-zero coefficient λ after the sign ≡ as it is relatively unimportant.
page 15 note * Invariants, p. 308. The proof is given in Proc. London Math. Soc., (2) 30 (1930).
page 15 note † Invariants, p. 54.
page 16 note * Invariants, p. 304.