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



From these equations, which contain a solution (for complex numbers) 

 of the problem solved for real numbers in art. 250 of the * Disquisitiones 

 Arithmeticse/ we may nifer the following theorems (Disq. Arith. art. 251 

 and 253) :— 



" The number w of classes of any order 12 is a divisor of the number 

 n of uneven classes of the same determinant D ; and, given any two classes 



of order O, there are always - uneven classes which compounded with one 



0) 



of them produce the other." 



« If D= 1, mod 2, and if the classes of O are derived from semieven or 

 even primitives, w is a divisor of the number of semieven classes of 

 determinant D ; and, given any two classes of order €1, there are always 



— semieven classes which compounded with one of them produce 



w 



times the other." 



"If D^±l, mod 4, and if the classes of €1 are derived from even 

 primitives, w is a divisor of the number n" of even classes of determi- 

 nant D ; and, given any two classes of order li, there are alwavs — even 

 classes which compounded with one of them produce 2i times the other." 



III. Determination of the number of Ambiguous Classes. 



Any form (A, B, C), in which 2B=0, mod A, is called by Gauss an 

 ambiguous form ; but in the investigation which follows we shall for 

 brevity understand by an ambiguous form an uneven form of one of the 

 four types 



(i) (A, 0, C), 



(ii) ([1+OB, B, C), 



(iii) (2B, B, C), 



(iv) (2^B, B, C). 



To determine the number of uneven ambiguous classes of any determi- 

 nant D supposed not to be a square, we shall determine, first, the number 

 of ambiguous forms of determinant D ; and secondly the number of ambi- 

 guous forms in each ambiguous class. 



(1) Let II be the number of different uneven primes dividing D. The 

 number of ambiguous forms of the type (i) is 4 x 2'*, or 8 x 2'*, according 

 as D is, or is not, uneven. For we may resolve — D into any two rela- 

 tively prime factors, and may take one of them (with any sign we please) 

 for A, and the other for 0. There are no ambiguous forms of the 

 type (ii), unless D=z, mod 2, or =0, mod (1 +2)\ For in the equation 

 D=B (B — [l+^] C), if B is uneven, we have D=/, mod 2, because C 

 must be uneven; if B is semieven or even, we have D=0, mod (1-f z)^. 

 If D=i, mod 2, we resolve D into any two relatively prime factors X and 



Y, and writing B=X, B-(l -\-i) C=Y, we find C=^— ^, which is in- 



