198 Prof. H. J. S. Smith on Quadratic Forms ^c. [Dec. 5, 



dratic forms ; but we shall regard the infinite number of forms, which 

 have the same invariants, as corresponding, in the general theory, to the 

 infinite num»ber of forms which have the same determinant, in the theory of 

 binary quadratic forms. 



If the index of inertia of the form / is i. e. if can be transformed 

 by a substitution of which the coefficients are real into a sum of k positive 

 andw — ^negative squares, we attribute to the invariant the sign — , and 

 to every other invariant the sign + . Thus the numbers Vi. Va^ • • • 

 all positive; V;t+i' ... V ^"^^ alternately negative and positive, so 



that the discriminant is of the same sign as ( — 1)"-^, as it ought to be. 

 This convention with respect to the signs of the invariants will enable us 

 to comprehend in the same formulse the theory of the generic characters 

 of forms of any index of inertia. We shall, however, suppose that the 

 index of inertia is at least 1, i. e. we shall exclude negative definite 

 forms. The invariants of a positive definite form are all positive ; and 

 the index of inertia of any indefinite form, of which the invariants are 

 given, is always indicated by the ordinal index of its negative invariant. 

 We shall represent by D the product — I^x — I3X — I5X . . . , the last 

 factor being — or — I,^_2J according as n is even or uneven. 



If ^iz=~ fi^ the forms 6^ 9^, ^3, . . . O^.^ are the primitive concomitants, 



and the last the primitive contravariant, of or 0^ ; each one of them is 

 either uneven, i. e. properly primitive, or even, i. e. im.properly primitive. 

 Two forms, which have the same invariants, are said to belong to the same 

 order when the corresponding primitive concomitants of the two forms are 

 alike uneven or alike even. Y/hen the invariants are all uneven, and the 

 number of the indeterminates is also uneven, there is but one order, none 

 of the primitive concomitants being in this case even. Again, when the 

 invariants are all uneven, and the number n of the indeterminates is even, 

 there is either one order or two, according as D eee — 1, or = + 1, mod 4 ; 

 for in both cases there is an order in which all the primitive concomi- 

 tants are uneven, and in the latter case, besides this uneven order, there 

 is an even order, in which these forms are alternately even and uneven, the 

 two extreme forms 0^ and being even. In the general case, when the 

 invariants have any values even or uneven, if I. is even, ^. cannot be even ; 

 again, if I^- is one of a sequence of an even number of uneven invariants, 

 preceded and folio v;ed by even invariants, cannot be even. But if there 

 be a sequence of an uneven number of uneven invariants I^, '^i^i, . . Ij+2yj 

 preceded and followed by even invariants, the sequence of primitive con- 

 comitants 0^-, O.^p . . . 0j-_^2^. are all uneven if ^. is uneven, and are alter- 

 nately even and uneven if 0^ is even ; a secfdence of forms or invariants may 

 consist of a single form or invariant. We attribute the value to the sym- 

 bols lo and I^, the value 1 to the symbols and fi^ ; thus the invariant 



