1864.] Prof. Smith on Complex Binary Quadratic Forms* 381 



Table III. 

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



1=0, 1 

 1 = 2 

 1^2 



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



1=0, 1 

 1=2 

 I>2 



■ST, y 

 y 



•m, y 



(ui) D=iPS^ P^l, mod 4. 



1=0 

 1 = 1, 2 

 1>2 



TO", 66 



(iv) D=2PS2, P=l^-2^, mod 4. 



1=0 

 1=1, 2 

 1>2 



OT, ay 

 ts-, a, 7 

 tzr, a, y 



(v) D=(1+0PS^ P=l, mod 4. 



1=0 

 1=1 

 1>1 



y 



or, y, a 

 y, a, /5. 



(7 



or, a, /3. 



ff, y 

 ff, y, /3. 



a 

 a 



a, /3. 



0-, y 

 (T, y, a. 



(vi) D=(H-OPS^ P=l + 22, mod 4. 



1=0 

 1=1 



CT, /3, y 



1>1 ^<r, ft y 

 (vii) D=2(1+0PS^ P=l, mod 4. 



(T 



(7, a. 



1 = 

 1=1 

 1>1 



rsr, a/3 

 w, a, /3 



(T 



(7, y 

 y. 



(viii) D=i(l+OPS', P=l + 2i, mod 4. 



1 = 

 1=1 

 I>1 



cr, aj3y 

 a/3, y 

 a, /3, y 



The characters preceding the vertical line by which the Table is divided 

 are not independent, but are subject to the condition (arising from the 

 laws of quadratic residues) that their product must be a positive unit. To 



show that this is so, let D=i* (1 PS^ where a! and /3' are each 

 either or 1 ; also let y'=0, or 1, according as P=l, or =1+22, mod 4. 

 If m is a number prime to (1 -|-t)D and capable of primitive representation* 



* If m = ax'^-\-1hxy-\-c'ip, the representation of m by («, h, c) is said to be primitive 

 when the values of the indeterminates are relatively prime. 



