292 Prof. Smith on Complex Binary Quadratic Forms. [June 16, 



is one half of the number of uneven ambiguous classes, according as the 

 classes (<^o) and (^j) are identical or not ; i. e., according as the whole 

 number of semieven classes is equal to or is one-half of the whole number 

 of even classes. 



The demonstration in the * Disquisitiones Arithmeticae/ that the number 

 of genera of uneven forms of any determinant cannot exceed the number 

 of uneven ambiguous classes of the same determinant, may be transferred 

 without change to the complex theory. We thus obtain a proof (inde- 

 pendent of the law of quadratic reciprocity and of the theorems which 

 determine the quadratic characters of ^ and 1+0 impossibility of 



one-half of the whole number of assignable generic characters ; and from 

 that impossibility, as we shall now show, the quadratic theorems are them- 

 selves deducible. 



(1) If p is an uneven prime =1, mod 2, there are two genera of un- 

 even forms of determinant p : of these one is the principal genus, and has 



the complete characters ^'^^ = 1, y = l; the other, containing the form 



(z, 0, has the particular character y= — 1 ; whence it follows that 



every uneven form of determinant which has the character y=-f-l, is 



a form of the principal genus, and has the character J = + 1 . Again, 

 fp— l,mod4, the form ^2 i, uneven form of determinant^; 



this form has the particular character y = — 1, because — ^ i, mod 2 ; 

 it is therefore not a form of the principal genus ; but it has the character 

 ^^^ = 1, because 2i is a square ; therefore, if ^^1, mod 4, every uneven 



form of determinant p has the character j^i^ J = + 1 . 



(2) There is but one genus of forms of determinant i, and its complete 

 character is a= + 1 ; there is also but one genus of forms of determinant 

 1 +i, and its complete character is /3 = + 1. 



(3) Let p and q be uneven primes of which the imaginary parts are 



even ; to prove the law of reciprocity, it will suffice to show that if j^^ =1, 



then j^^J = 1 . The equation = 1 implies the existence of a con- 

 gruence of the type w^— ^=0, mod q, and consequently of an uneven 

 form of determinant^, and of the type^y, fa;,^^~^^. This form has the 

 character y= + l, because 2=1, mod 2; it therefore has the character 



'■•■[,]=■■ 



(4) To prove the equatloii j^lj = (~ 1)^ ''^^^ ^\ in which we may sup- 



