1864.] Prof. Smith on Complex Binary Quadratic Forms. 297 



taneously divisible by 1 + i, because (Z) is congruous, for the modukis 1 + i, 

 to the first or second of the matrices 



/O, I, 1, 0\ , (1, 0, 0, 1\ .2'^ 



Uo.o.ij ''"Ho, i,i,o> .(^^ 



according as a=i, c=\, or a = l, c=i, mod 2. Consequently is a 

 form the duplication of which produces /. When D=l, or=0, mod 2, 

 let the prime matrix 



oc", (3" 



transform —x^—y^—z^ into («, —5, e). As we cannot have simultaneously 

 a=/3, a'=/3', a"=/3", mod (l+^), we , may suppose that or and /3 are 

 incongruous, mod (1+0- If ^ = (B + 2C, ^A, B— ^■C), the matrix 



//3'+i/3", i/3, /3^-t/3"\ 



\a' + za", za, za, a' — eV/ 

 transforms / into ^ X ^, and is a prime matrix, being congruous to one or 

 other of the matrices (Z') for the modulus l+z, in consequence of the 

 two suppositions that a and c are uneven, and that a and /3 are incongruous, 

 mod ( 1 + z) : so that / arises from the duplication of ^. 



From the resolubility of this problem we can infer (precisely as Gauss 

 has done in the real theory) that that half of the assignable generic cha- 

 racters which is not impossible corresponds to actually existing genera. 

 We can also deduce a demonstration of the theorem that any form of de- 

 terminant D can be transformed into any other form of the same genus, 

 by a transformation of which the coefficients are rational fractions having 

 denominators prime to 2D. For every form which arises from the dupli- 

 cation of an uneven primitive form — that is, every form of the principal 

 genus — represents square numbers prime to 2D, and is therefore equivalent 



^ \ But (1, 0, — D) is transformed 



into ( Hi ^ ^ ) I t ' ^* ^* forms of the principal genus 



can be transformed into one another by transformations of the kind indi- 

 cated. Again, if f^,/^ be two forms of any other genus, a form cp of the 

 principal genus exists satisfying the equation /2=0X/i. But since can 

 be transformed into the principal form, we can assign to the indeterminates 

 of (j> rational values, having denominators prime to 2D, which shall cause ({> 

 to acquire the value + 1 ; and thus, from the transformation of into/^ X 0, 

 we deduce a rational transformation of into/^, the coefficients of which 

 have denominators prime to 2D. The truth of the converse proposition, 

 " Two forms which are transformable into one another by rational trans- 

 formations having denominators prime to 2D belong to the same genus," 



