1864.] Prof. Smith on Complex Binary Quadratic Forms. 295 



pUes a transformation of any given form of determinant 1 into one or other 

 of those two forms. 



If D=2, or 1+z, mod 2, no binary form of determinant D can he re- 

 presented by —x^—y^—z^i because D cannot be represented by the con- 

 travariant of that form, z. e. by the form — — — itself. Conse- 

 quently, if D=2, or 1 +«, niod 2, the binary forms of its principal genus 

 are certainly capable of primitive representation by -f iy"^ + iz^. 



If D = l, mod 2, no form of the principal genus can be primitively 

 represented by •^ly'^ -\-iz^ . Let f—{a, h, c) be such a form, and let us 

 suppose, as vre may do, that b is even, so that ac=l, mod 2, and 

 «=c=l, mod 2 (the supposition a=c=z is admissible, because /is of 

 the principal genus) ; if possible, let the prime matrix 



a", /3" 



(of which A, B, C are the determinants) transform x^-^-iy'^-^iz'^ into f; we 

 have the equations «=aHza'H?V^ c=/3H^/3''+^/3"^ D= A'-eB'— zC^ 

 from which, and from the congruences T)=a=c=\, mod 2, we infer the 

 incompatible conditions a' + ?a''=/3' + «73"=0, modi +z, A=l, modi + 

 i.e. /is incapable of primitive representation hj x'^-\-iy^-\-iz^. If, there- 

 fore, D=l, mod 2, the forms of its principal genus are capable of primi- 

 tive representation by — x^—y'^—z^. We may add that when D= + l, 

 mod 4, the forms of that genus which diifers from the principal genus 

 only in having the character y= — 1, instead of y=-l-l, are capable of 

 primitive representation by a:;^ + zy^ + but not by —x^—y'^—z'^. 



Lastly, let D=0, mod 2. If D=2, or=2(l +z), mod 4, D cannot be 

 primitively represented by o?^ — zy^ — zV, the contravariant of x"^ -{-iy"^ -\-iz'^ \ 

 i.e. no form of determinant D can be primitively represented by a?^+zy^+z2'^; 

 so that forms of the principal genus are certainly capable of primitive repre- 

 sentation by —x^—y'^ — z'^. But if D = 2z, or =0, mod 4, the forms of the 

 principal genus are capable of primitive representation by both the ternary 

 forms —x^—y^—z^ and .r^ + zy + z>^. For if /=(«, b, c) be a form of the 

 principal genus of any even determinant,/ can only represent numbers=0, 

 or = l, mod 2 ; so that a ternary form of determinant 1 and of the type 



f+p"z^+2qyz-\-2q'xz 

 will be equivalent to —x^—y^—z^, or to ^^ + zy+z>^, according as^"=0, 

 or = l, mod 2, on the one hand, or^"=z, or = 14-«> on the other hand. 

 Again, if {It, k!) is a value of the expression ^ {a, —b, c), mod D, (in 



which we now suppose a uneven and b semieven or even), (^"^ ^^i* 

 is another value of the same expression ; and it can be shown* that when 



^ f-\-]^"z'^-\-2qy'^-\-2q'xz is a ternary form of det. 1, derived from the value {7c, h') 



of the expression i^' {a, —b, c), mod D, k is the coefBcient of j/z in the contravariant 



a — 



form. Hence a=k^ —'D{q'^ —ap"), or ap"=q'^-\ — Observing thata=l, mod 2 



