298 REPORT — 1861. 



with regard to them, though it is not necessary to exclude them (as is the 

 case with forms of determinant zero) from those investigations which 

 relate simultaneously to the two remaining kinds of quadratic forms; viz. 

 (3) those of a negative determinant, and (4) those of a positive and not 

 square determinant. An essential difference between these two kinds of 

 forms is, that whereas both positive and negative numbers can be represented 

 by any form of positive and not square determinant, forms of a negative deter- 

 minant can represent either positive numbers only, or negative numbers only. 

 For if the roots of a+269+c9^=0 be real, it is clear that ax^ + 2bxy -\- cif 

 will have values of different signs, when the ratio y : x falls between the two 

 roots and when it falls outside them ; but if the roots be imaginary, the form 

 will always obtain values having the same sign (viz. that of a or c), whatever 

 the ratio y : x may be. If (a, b, c) be a positive form (i. e. a form repre- 

 senting positive numbers only) of a negative determinant D=— A, 

 (_a, _6, _c) is a negative form of the same determinant, and can repre- 

 sent negative numbers only. We see, therefore, that there are as many 

 positive as negative classes for any negative determinant; and as everything 

 that can be said about positive forms or classes may be transferred at once, 

 mutatis mutandis, to negative forms and classes, we shall in what follows 

 exclude the latter from consideration, and, when we are speaking of forms of 

 a negative determinant, confine ourselves to the positive forms. 



Sincea;^ — D3/^or(l, 0,— D),is a form of determinant D, we see that one class 

 at least of properly primitive forms exists for every determinant ; and the class 

 containing the form x'^ — Dy^ is called the principal class. Improperly pri- 

 mitive forms only exist for those determinants which satisfy the condition 

 D=l, mod 4 ; since, if (a, b, c) be improperly primitive, we have b~l, mod 2, 

 a^c^O, mod 2. But for every determinant satisfying this condition, one 



class at least of improperly primitive forms exists; for (2, 1, -— j 



improperly primitive form of determinant D, and the class containing it may 

 be called the principal class of improperly primitive forms. 



86. Reduction of the Problem of Representation to that of Equivalence. — 

 The problem of the representation of numbers depends, first, on the solution 

 of a quadratic congruence, and, secondly, on the solution of a problem of 

 equivalence. This dependence is established by the two following theorems: — 



(i.) " When the number M admits of a primitive representation by (a, b, c), 

 the quadratic congruence a.' — D=0, mod M, is resoluble." 



For if am'^-H26m«+cw^=M be a primitive representation of M, let fx, v be 

 two numbers satisfying the equation mv—nfi=^V ; we then find 



(am^-\-2bmn+en'){aii'- + 2biiv + cv'') — (amix-\-b[mv+nii']-\-cnvy—'D-, 



or ii'^=D, mod M; i^ Q.-=amii-\-b{mv+nfi'\+cnv. 



We have already referred to this result in art. 68. 



The representation am"^ -'r2bmn + cn- of the number M by the form (a, b, c), 

 is said by Gauss to appertain to the value Q, of the congruential radical 

 ■v/D, mod M. To understand this definition with precision, it is to be 

 observed that if in the expression of Q, we replace /i and y by any two other 

 numbers satisfying the equation my — n/x^l, the new value of ii will be of 

 the forra^-1-AM; and conversely, values for n and v can always be found 

 which shall give to amfi-{-b\_mv + n^'] +cni' any assigned value of the form 

 Oi + kM. Two different representations of M appertaining to the same value 

 of \/D, mod M, are said to belong to the same set. 



IS an 



