III. On the orders and genera of quadratic forms containing more than three indeterminates.—Second notice

Abstract

The principles upon which quadratic forms are distributed into orders and genera have been indicated in a former notice (Proceedings of the Royal Society, vol. xiii. p. 199). Some further results relating to the same subject are contained in the present communication. I. The Definition of the Orders and Genera. Retaining, with some exceptions to which we shall now direct attention, the notation and nomenclature of the former notice, we represent by f 1 a primitive quadratic form containing nindeterminates, of which the matrix is || A n x n i, j ; by f 2 , f 3 , . . . f n -1 , the fundamental concomitants o f 1 , of which the last is the contravariant. The matrices of these concomitants are the matrices derived from the matrix of f 1 , so that the first coefficients of f 2 , f 3 , .. . f n -1 , are respectively the determinants |A 2 x 2 i, j |, | · A 3 x 3 i, j |,... |A n -1 x n -1 i, j |, taken with their proper signs. The discriminant of f 1 , i. e. the determinant of the matrix |A n x n i, j |, which is supposed to be different from zero, and which is to be taken with its proper sign, is represented by ∇ n . The greatest common divisors of the minors of the orders n - 1, n - 2, . . . 2, 1 in the same matrix are denoted by ∇ n -1 , ∇ n -2 , ∇ 2 , ∇ 1 , of which the last is a unit; we shall presently attribute signs to each of these greatest common divisors.