1864.] Prof. Smith on Complex Binary Quadratic Forms. 283 



w " Ml is the least common muUiple of and (Gauss's 5th 



M3 m 3 



and 6th conchisions, Disq. Arith. art. 235.) 



It will be found that Gauss's proof of these theorems can be transferred to 

 the complex theory ; only, when / and f are both semieven, or derived 

 from semieven primitives, the proof of the sixth conclusion is incomplete, 

 and, ^Yhile showing that F cannot be derived from an uneven primitive, 

 fails to show whether it is derived from a semieven or from an even primi- 

 tive. But, in the same way in which Gauss has shown that is divisible 

 hy m^ X ni\, it can also be shown that is divisible by m.^ x m'^* ; i. 

 in the case which we are considering, M., is divisible by M^, because 

 rii.^=zm^y m'^=m\, and m^m\=M-^. Therefore M^^M^ and F is derived 

 from a semieven primitive in accordance with our enunciation of Gauss's 

 sixth conclusion. 



(2) In the real theory, when two or more forms are compounded, each 

 form may be taken either directly or inversely ; but, however the forms 

 are taken, the determinant of the resulting form is the same. In the 

 complex theory, not only may each of the forms to be compounded be 

 taken in either of two different ways, but also the determinant of the re- 

 sulting form may receive either of two values, differing, however, only in 



sign ; and it is important to attend to the ambiguities which thus arise. 



p 



If a complex rational number n be written in the form where 



Q 



X is 0, 1, 2, or 3, fx is any positive or negative integer, and P, Q are primary 

 uneven complex integers, we may term the sign of n. Let F, of which 

 the determinant is D, be transformed into the product x/2 X ... /i, by 

 a substitution [X, Y] linear and homogeneous in respect of h binary sets ; 

 we have, as in the real theorv, 7i equations of the type 



\clv^ dij^ dy^ d.v ) B fk ' 

 representing the determinant of . Let 



so that tfj^ =^ ; if i^^ is the sign of , we shall say that fj^ is taken with 



the sign i"^. We can thus enunciate the theorem, "Forms, compounded 

 of the same forms, taken with the same signs, are equivalent." If 



. . . /a are given forms which it is required to compound, the signs of 

 d^y d^, . . .dh must be all real, or else all unreal ; and the sign of D will be 

 real or unreal accordingly. The value of D (irrespective of its sign) is 

 ascertained as in the real theory ; but it may receive at our option, in the 



* Disq. Arith. art. 235. The proof that 2{hb'+A) and 2{bb'-A) are divisible by 

 miXfn\, may be employed {mutatis mictandis) to show that (1 ■+•?) (ii' + A) and (1+0 

 (bb' — A) are divisible by X m\^. 



