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

 by/, the congruence ur=J), mod m, is resoluble ; and its resolubility im- 

 plies the condition |^^J = x [^^-^^J >^ [^] = 

 laws of quadratic residues, [^^J j^^— =A t"] t^] ' 



the condition just written becomes a" (f l^^]"''' which is coinci- 

 dent with that indicated in the Table. Thus (as in the real theory) one- 

 half of the whole number of assignable generic characters are impossible * ; 

 we shall presently obtain a different proof of this result, and shall also 

 show that the remaining half correspond to actually existing genera. 

 For the characters of a semieven form f, it is convenient to take the 



characters of the numbers represented by -^-j-. \ ^o^* characters of 

 an even form, the characters of the numbers represented by The 



following Table will serve to form the complete generic character in each 

 case. 



For a semieven form, 

 (i) D=PS% P=l, mod 4. 

 1 = \ vs\a. 



(ii) D=PS2, P=l + 2^, mod 4. 

 1=0 \'us,y\a. 



For an even form. 

 1=0 \ -m\fy. 



II. The Theory of Composition. 



The theory of composition given in the * Disquisitiones Arithmeticse ' 

 is immediately applicable to complex quadratic forms. There are, how- 

 ever, a few points to which we must direct attention. 



(1 ) If m^, are the greatest common divisors of a, 26, c ; a, (\-\- i)b, 

 c ; a, h, c, we have 



(i) m^=m2=7)i3, 



(ii) m^=m^ = (l+i)m^, 



(iii) = ( 1 + i)m^ = ( 1 + iy'>^h> 



according as (a, h, c) either is, or is derived from, (i) an uneven, (ii) a semi- 

 even, (iii) an even primitive. Hence the order of a form is given when 

 and are given. Thus, if F is compounded of / and/, and if M2 M3, 

 W3, m\ m\ m\ refer to F, /, / respectively, the order of F is com- 

 pletely determined by the two theorems, " is the product of and 

 * The determinant is supposed not to be a square. 



