ON THE THEORY OF NUMBERS. 323 



divide D, the total number of generic characters that can be formed by com- 

 bining the particular characters in every possible way is S** whenD^lorS, 



mod 8; '^ when D^O, mod 8; and S**"*"' in every other case. But it 

 follows from the law of quadratic reciprocity, that one-half of these complete 

 characters are impossible ; i. e. that no quadratic form characterized by them 

 can exist. To see this, we observe that if m be a positive and uneven num- 

 ber prime to D, and capable of primitive representation by^J the congruence 



^^— D^O, modm, is resoluble; and consequently { _ ) =-f 1. Therefore 



, /P\ /2P\ W 



also J — j = l, or ( — j=l, according as D is of the form PS' or 2PS'. 



In the first case we have (^ ) =(-l)*^'"~^^^^"'\ or 



in the other owe {^)i^)=l: i- e. (p)=(-I)»<-«''-''*^, or 



(2)(^)...=(-,)«-x-«^. 



A comparison of these equations with the preceding Table will show that the 

 product of the particular characters which stand before the line of division 

 in the Table is equal to +1 in the case of any really existing genus; i.e. that 

 precisely one-half of the whole number of complete generic characters are 

 impossible. We shall hereafter see that the remaining half of the generic 

 characters correspond to actually existing genera, and that each genus con- 

 tains an equal number of classes. That genus, every particular character of 

 which is a positive unit, is called the principal genus ; it evidently contains 

 the principal class, and is therefore, in every case, an actually existing 

 genus. 



Since the extreme coefficients of a form are numbers represented by it, 

 and since, further, if the form be properly primitive, one or other of them is 

 prime to 2 and to any prime divisor of the determinant, we see that the 

 generic character of a form can always be ascertained by considering the 

 values of its first and last coefficients. Thus the complete character of the 

 form (11, 2, 15), of which the det. is —161 = — 7x23 (case II. (a) in the 



Table), is (^)=1, (^) = ~1' •(-1)'^ = -1; that of (5, 2, 33) is 



Two forms, which have different generic chamcters, cannot be equivalent ; 

 nor can a number be represented by a form if its character is incompatible 

 with the generic character of the form. It is therefore convenient, in any 

 problem of equivalence or representation, to begin by comparing the generic 

 characters of the given forms with one another, or with the characters of the 

 given numbers. 



The uneven numbers prime to the determinant, which are represented by 

 forms of the same genus, are contained in one or other of a certain number 

 of linear forms. If R denote the product of the primes r, r', . . already de- 

 fined, and if be any term of a system of residues prime to 2*PR, where k 

 is 1, when D ^ 1 or 5, mod 8, 3 when D ^ 2, 6, or 0, and 2 in every other 

 case, the numbers contained in the formula 2^PR -}- can be represented only 



y2 



