324 REPORT — 1861. 



bv forms belonging to that genus the character of which coincides with the 

 character of the number 0. It is clear that one half of tlie linear forms, in- 

 cluded in the formula 2''PR + 0, do not satisfy the condition of possibility in- 

 dicated in the Table, and are therefore incompatible with any quadratic form 

 of determinant D ; while the remaining half of those linear forms will be 

 equally distributed among the actually existing genera; so that there will be 

 either'n^(p—l)j(r—'[)or2U^(p—l)^(r—l) linear forms proper to each 

 genus. But although no number contained in any one of the first-named 

 linear forms can be represented by a form of determinant D, yet it is not to 

 be inferred that every number m contained in the other half of the linear 

 forms is capable of such representation ; for from the linear forio of m, we 



can indeed infer the equation ( _ )= 1 ; but, if m be not a prime, or at least 



the product of a prime by a square, we cannot from this equation infer the 

 resolubility of the congruence i2^ 5s D, mod m, or of any congruence of the 



forraii^^D, mod—. We may add- that if we assume the theorem that 



every arithmetic progression, the terms of which are prime to their common 

 difference, contains prime numbers, the consideration of the case in which tn 

 is a prime establishes the actual existence of every genus the character of 

 which satisfies the condition of possibility. (Crelle, vol. xviii. p. 269.) 



If 7>i be an uneven number not divisible by q, a prime divisor of D, and if 

 the double of m can be represented by an improperly primitive form^of 



det. D, we attribute to/ the particular character | — j = +1, or = — 1, 

 according as ( — ) = + 1, or = — 1 ; and to form the complete character off, 



we may use the Table 



D=PS'-, P = l, mod4, S = l, mod2. 



(■^)' {^- I (^)' {^■■■■' 



99. In the preceding articles we have briefly recapitulated the definitions 

 and principles which constitute the elements of the theory of quadratic forms. 

 We have hitherto followed closely the 5th section of the Disq. Arith. (arts. 

 153-222 and 223-233) ; but before we proceed to an examination of the re- 

 mainder of that section, it will be convenient to place before the reader an 

 account of the method employed by Lejeune Dirichlet in his great memoir, 

 " Recherches sur diverses applications de I'analyse infinitesimale a la theorie 

 des nombres," for the determination of the number of quadratic forms of a 

 given positive or negative determinant. 



* All the results of this article are given in the Disq. Arith. arts. 223-232 ; but as Gauss 

 does not employ the symbol of reciprocity, we have preferred to follow the notation of Di- 

 richlet. It is also to be noticed that Gauss does not use the law of quadratic reciprocity to 

 demonstrate the impossibility of one-half of the generic characters ; for, as we shall here- 

 after see, this impossibility is proved in the Disq. Arith. (art. 261) independently of the law 

 of reciprocity, and is then employed to establish that law. (Gauss's second demonstration, 

 see Disq. Arith. art. 262.) There is also an unimportant difference between Dirichlet and 

 Gauss with respect to the definition of the generic character of an improperly primitive form ; 

 for Gauss obtains the generic character (see art. 232) by considering the numbers repre- 

 sented by the form, and not the halves of those numbers. But he also observes (art. 227, 

 and 250, VI.) that each improperly primitive class is connected in a particular manner (to 

 which we shall again refer) with one or with three properly primitive classes ; and that this 

 consideration may be employed to divide the improperly primitive classes into genera. And 

 it will be found that the complete character which Dirichlet's definition attributes to an im- 

 properly piimitive form is, in fact, the complete character of the properly primitive class or 

 classes with wluch it is connected. 



