1864.] Prof. Smith on Complex Binary Quadratic Foims, 287 



tegral because X and Y are uneven, and uneven because X is not =Y, 

 mod 2. Thus if D = f, mod 2, there are 4x2'* ambiguous forms of the 

 type (ii). Again, if D = 0, mod (l-\-iy, we may resolve D in any way 

 we please into two factors having l-\-i for their greatest common divisor ; 

 we find in this way 8x2'* ambiguous forms of the type (ii). There 

 are no ambiguous forms of the types (iii) or (iv), unless I) = l, mod 2, or 

 = 2, mod 4, or =0, mod (l+O'- For if in the equation D = B(B — 2C), we 

 suppose B uneven, we find D=l, mod 2 ; if B is semieven, W=2i, and 

 2BC=2(1 +z), mod 4, whence D=2, mod 4 ; lastly, if B is even, D=0, 

 mod {\-\-iy. The same reasoning applies to the equation D = B(B— 2zC). 

 If mod 2, we resolve D in every possible way into the product of 



two factors relatively prime; let D = XxY be such a resolution, then 

 D = ?-Xx — zY is another ; and it will be seen that according as the last 

 coefficient in the two forms 



is uneven or not uneven, so the last coefficient in the two forms 



[2,xa-,i^+iX], [-2X.-x.^]. 



is not, or is, uneven ; i. e. there are 2x2'* ambiguous forms of each of the 

 types (iii) and (iv). If D=2, mod 4, we resolve D in every possible 

 way into two factors, of which \-{-i is the greatest common divisor ; we 

 thus find 4 X 2m uneven forms of each of the types (iii) and (iv) . Lastly, 

 if D=0, mod (l+z)', we resolve D in every possible way into two factors 

 of which 1 +2 is the greatest common divisor, and we obtain 8x2 forms 

 of each of the types (iii) and (iv). 



The result of this enumeration is that if D be uneven, or semieven, or 

 = 2z, mod 4, there are 8x2'* ambiguous forms ; if D=2, mod 4, or =0, 

 mod {l+ify but not mod (1 +z)', there are 16x2'*; and if D = 0, mod 

 (1 +iy, there are 32 x 2'*. On comparing this result with Table III., it 

 will be seen that in every case there are four times as many ambiguous forms 

 as there are assignable generic characters. 



(2) Let /=(«, h, c) be any form of an ambiguous class ; if (I) = j^'~^| 

 is an improper automorphic of /, X, fi, v satisfy the equations 



^^-X.= l, . , (1) 



Xa + 2ju6 + )^c=0 ; (2) 



and, conversely, if X, yu, v satisfy the equations (1) and (2), (I)= ^'""^ | 



is an improper automorphic of /. Let a, y, p, q (of which a and y are 

 relatively prime) be a system of integral numbers satisfying the equations 



pCC = X, ^y=^-l, I 



