1867.] Prof. H. J. S. Smith on Quadratic Forms ^c. 203 



The converse is true, subject to certain hmitations : — 



" If M is prime to and not negative except when is negative, 

 and if — I„_iX0„_2 is a quadratic residue of M, (ji^ is cipable of primitive 

 representation by /i." 



*' If, in addition, M is prime to I;,_2, there is ahvays either one or two 

 genera of forms of the invariants (Ii, I25 • • • M L„_2) capable of primitive 

 representation by forms of a given genus of the invariants (Ii, L, . . . I„_2, 

 I„_i) ; and if there are two genera capable of such representation, they 

 are of different orders." 



These theorems are especially useful in the theory of definite forms, to 

 which, for the remainder of this paper, we shall confine our attention. In 

 the case of such forms we understand by the weight of a form the reciprocal 

 of the number of its positive automorphics, by the weight of a class the 

 weight of any foi'm representing the class ; the weight of a genus, or 

 order, is the sum of the weights of the classes contained in the genus or 

 order ; the weight of a representation of a number by a form is the weight 

 of the representing form ; the weight of a representation of a form by a 

 form is the product of the weights of the representing and represented 

 forms. 



Let r denote a system of forms, representatives of a given genus of the 

 invariants T2, » . • I«-i ; iet M be a number divisible by fi different 

 uneven primes, none of which divide any of the invariants, and let M be 

 uneven or unevenly even, according as the contravariants of the forms V 

 are uneven or even ; we tben have the theorem — 



" The sum of the weights of the representations of M by the contra- 

 variants of the forms r, is 2^^ times the weight of the single genus, or the 

 two genera, of invariants I^, lo, . . . MI„_2, which admit of representation 

 by the forms r." 



The method which this theorem may serve to indicate supplies a solu- 

 tion of the problem " to determine the weight of a given genus of definite 

 forms of n indeterminates, and of the invariants I^, I2, . . . I„_i ..." We 

 shall represent the weight of the given genus by the formula 



W=4 X n . x(3) X x'n'l/''^'^-^> 



s=l 



v,hen n is uneven and equal to 2j' + 1, and by the formula 



w=f„. X n . x(0 X B., x'=r'l/"<»-'> x\s(^)L 



s~i 1 \ 7U / 



when n is even and equal to ; and we sball consider separately the 

 factors of which these formulae are composed. 



GO 



(i) In the infinite series — ^ — (which enters into the expression 

 of W only when the number of indeterminates is even) D still represents the 



