The purpose of this note is to extend to complex quadratic forms some important investigations of Gauss relating to real quadratic forms. We shall consider in order (I.) the definition of the Genera, (II.) the theory of Composition, (III.) the determination of the number of Ambiguous Classes, (IV .) the representation of forms of the principal genus by ternary quadratic forms of determinant 1. For the comparison of the numbers of classes of different orders, we may refer to a paper by M. Lipschitz (Crelle’s Journal, vol. liv. p. 193) ; and for the principles of the theory of complex numbers and complex quadratic forms, to Lcjeune Dirichlet’s Memoir, “ Recherches sur les formes quadratiques à coefficients et à indéterminées complexes ” (Crelle, vol. xxiv. p. 291). I. The Definition of the Genera . Let f =( a, b, c ) be an uneven primitive form of determinant D, and m = ax 2 + 2 bxy + cy 2 , m' = ax’ 2 + 2 bx'y' + cy' 2 two numbers represented by f . The generic characters of f are deducible from the equation ( ax 2 + 2 bxy + cy 2 ) ( ax’ 2 + 2 bx'y' + cy' 2 ) = ( axx’ + b [ xy' + x'y ] + cyy' ) 2 - D ( xy' - x'y ) 2 , or, as we shall write it, mm' = P 2 - DQ 2 .