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



way as the unit (i, i') : we may omit, however, from the exponent of 

 — 1 in its expression every term into which an even form enters ; if, for 

 example, 0^ is an even form, that exponent contains the terms 



and no other term into which 6^ enters; but ^'§ = 0, and the coefficient of 

 4 (0,— 1 ) is even ; so that 0^ disappears from the expression of the unit 

 ■Ij (0, n—\). It will thus be seen that the equation (A) involves only 

 generic characters (principal, supplementary, or simultaneous) of the con- 

 comitant system : that equation therefore expresses a relation which the 

 complete character must satisfy. 



In using these formulae, we must attend to the significations which we 

 have assigned to the symbols lo, I„, Oo» ^"cl 0^^. Thus 



00-1 00-1 

 (-1)"2 =l=(-l)-y-, f,^ >3, etc. 



We shall conclude this part of our subject with the two theorems : — 



(i) Every genus, of which the character satisfies the condition of pos- 

 sibilit}^, actually exists. 



(ii) Twp forms, of the same invariants, of the same order, and of the 

 same genus are transformable, each into the other, by rational Imear sub- 

 stitutions of which the determinants are units, and in which the denomi- 

 nators of the coefficients are prime to any given number. 



The first of these theorems shows that the condition of possibility is suf- 

 ficient as well as necessary ; the second establishes the completeness of the 

 enumeration of ordinal and generic ciiaracters. 



II. Determination of the Weight of a given Genus of Definite Forms. 



It has been shown by Gauss, in the digression on ternary forms in the 

 fifth section of the ' Disquisitiones Arithmeticse,' that the solution of the 

 problem.s " to obtain all the representations of a given binary form, or of a 

 given number, by a given ternary form," depends on the solution of the 

 problem *'to determine whether two given ternary forms are equivalent, 

 and, if they are, to obtain all the transformations of either of them into the 

 other." Similarly the solution of the problem ''to obtain all the repre- 

 sentations of a given quadratic form of i indeterminates (2= 1, 2, . . . n — 1) 

 by a given form of n indeterminates " depends on the solution of the pro- 

 blem of (equivalence for quadratic forms of n indeterminates. The follow- 

 ing proposition is here of primary importance : — • 



"If the form ofn—X indeterminates and of the invariants I,, T2, . . . 

 I,j_3> MI^_2, is capable of primitive representation by the form 0^ of n 

 indeterminates, and of the invariants I^^, I^, . . . I,^_.3, I^_2, then 

 — ln-\ X(Pn-2 (^''here is the primitive contravariant of 0j is a qua- 

 dratic residue of M." 



