1867.] Prof. H. J. S. Smith on Quadratic Forms ^c. 197 



III. '^On the Orders and Genera of Quadratic Forms containing 

 more than three Indeterminates.^^ By H. J. Stephen Smith, 

 M.A., F.U.S., SaviUan Professor of Geometry in the University 

 of Oxford. — Second Notice. Received October 30, 1867. 



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 sub- 

 ject 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 a 

 primitive quadratic form containing n indeterminates, of which the matrix is 



|l ''^i> j li ; /a'/s' ■ • 'fn-\^ the fundamental concomitants of f, of which the 

 last is the contravariant. The matrices of these concomitants are the ma- 

 trices derived from the matrix of f, so that the first coefficients of f^^f, . . . 



2X2 3X3. |«-lXn-l| 



fn-iy are respectively the determmants A^, ^1, • Aj,y|, . . . | A,-, • |, taken 

 with their proper signs. The discriminant of i. e. the determinant of 

 the matrix | A^, I, which is supposed to be different from zero, and which 

 is to be taken with its proper sign, is represented by v,i« The greatest 

 common divisors of the minors of the orders n — \, n—2, ... 2, 1 in the 

 same matrix are denoted by V„._p V„_2> • • Vp of which the last is a 

 unit ; we shall presently attribute signs to each of these greatest common 

 divisors. The quotients 



"^w-l Vn-2 Vn-3 Vi 1 



which are always integral, we represent by . . . Ii ; so that 



The numbers 1^ I^, . . are the first, second, .... last invariants of the 

 form f, and remain unchanged when/^ is transformed by any substitution 

 of which the determinant is unity and the coefficients integral numbers. 

 Forms which have the same invariants have of course the same discrimi- 

 nant ; but (if the number of indeterminates is greater than two) forms which 

 have the same discriminant do not necessarily have the same invariants ; 

 for example, the quaternary forms 



+ < 4- 2Ay + 6.v/, .v^^ -\- -|- ay + 1 2 ay 

 have the same ciiscrmiuant 12, but their invariants 1^ I^, are respectively 

 1, 2, 3, and 1, 1, 12. As forms which have the same discriminant, but 

 different invariants, do not necessarily have any close relation to one 

 another, we shall not employ the discriminant in the classification of qua- 



