XII. On the orders and genera of ternary quadratic forms

Abstract

Eisenstein, in a Memoir entitled "Neue Theoreme der höheren Arithmetik", has defined the ordinal and generic characters of ternary quadratic forms of an uneven determinant; and, in the case of definite forms, has assigned the weight of any given order or genus. But he has not considered forms of an even determinant, neither has he given any demonstrations of his results. To supply these omissions, and so far to complete the work of Eisenstein, is the object of the present memoir. Art. 2. We represent by f the ternary quadratic form a x 2 + a ' y 2 + a '' z 2 +2 byz +2 b ' xz +2 b '' xy ; . . . . . . . (1) we suppose that f is primitive ( i. e . that the six integral numbers a , a ', a '', b , b ', b " admit of no common divisor other than unity), and that its discriminant is different from zero; this discriminant, or the determinant of the matrix a , b ", b ' b ", a ', b b ', b , a " . . . . . . . (2) we represent by D ; by Ω we denote the greatest common divisor of the minor determinants of the matrix (2); by Ω F the contravariant of f , or the form ( a ' a ''— b 2 ) x 2 + ( a " a — b ' 2 ) y 2 + ( a a '— b '' 2 ) z 2 + 2( b ' b "— ab ) yz + 2( b " b — a ' b ') zx + 2( bb ' — a '' b ") xy ; }. . . . . . . . (3) we shall term F the primitive contravariant of f , and we shall write F = A x 2 + A' y 2 + 2B yz + 2B' xy + 2B'' xy . . . . . . (4)