1864.] 



containing more than Three Indeterminates, 



201 



principle of arrangement is adopted in writing the quadratic matrix (A^^)), 

 and the transforming matrix (a<^^)) ; provided only that the arrangement 

 be the same in the two matrices, and that in each matrix it be the same in 

 height and in depth. 



For example, if f^= x\ + a^xl-\- x\ + a^x\ + 25^ x^ x^ + 25^ x^ + 

 x^ x^-\-2h^ x^ + 2h. x^ x^ + 2^6 x^ x^ be a quadratic form containing four 

 indeterminates, the form = 



+ 2(5, b,-a, bj X, X, + 2(5, b-a, b,) X, X, 

 -2(5, 5,-«, 5 J X, X, - 2(5, b-a, b,) X, X, 

 -2(5, 5,-53 h) X, X, + 2(5, 53-^, 5,) X, X3 

 + 2(5, 5,-«3 5,) X, X, - 2(5, 5^-53 5J X, X, 

 -2(5, 5,-«3 53) X, X3 - 2(5, 5,-5, 5,) X3 X, 

 + 2(53 b-a, 5,) X3 X, + 2(53 ^e-«. y X3 X3 

 + 2(5,5-^,5,) X,X, - 2(5,53-«3 5,) X, X, 

 + 2(5,5,-«,5,) X,X, 

 is the concomitant of the second species of /. 



The n—l forms defined by the formula (A), of which the first is the 

 form/, itself, and the last the contravariant of /,, we shall term the funda- 

 mental concomitants of f; in contradistinction to those other quadratic 

 concomitants (infinite in number) of which the matrices are the symme- 

 trical matrices that may be derived, by a multiplicate derivation, from 

 (A(i)) .... Passing to the arithmetical theory of quadratic forms — i. e. 

 supposing that the constituents of (ACO) are integral numbers, we shall 

 designate by Vi> V2' • • • Vn the greatest common divisors (taken posi- 

 tively) of the minors of different orders of the matrix (AO)), so that, in 

 particular, Vi is the greatest common divisor of its constituents, and Vn is 

 the absolute value of its determinant, here supposed to be different from 

 zero. By the primary divisor of a quadratic form we shall understand the 

 greatest common divisor of the coefficients of the squares and double rect- 

 angles in the quadratic form ; by the secondary divisor we shall understand 

 the greatest common divisor of the coefficients of the squares and of the 

 rectangles ; so that the primary divisor is equal to, or is half of, the 

 secondary divisor, according as the quadratic form (to use the phraseology 

 of Gauss) is derived from a form properly or improperly primitive. It 

 will be seen that Vi> Va* • • • • Vn-\ are the primary divisors of the forms 

 . . . respectively. 

 We now consider the totality of arithmetical quadratic forms, contain- 

 ing n indeterminates, and having a given index of inertia, and a given de- 

 terminant. 



If a quadratic form be reduced to a sum of squares by any linear trans- 

 formation, the number of positive and of negative squares is the same. 



