On indefinite ternary quadratic forms

In a paper [1] with the same title Barnes has shown that if Q(x, y, z) is an indefinite ternary quadratic form of determinant d ≠ 0 then there exist integers x1, y1, z1, x2,···z3 satisfying for which Furthermore, unless Q is equivalent to a multiple of or two other forms Q2, Q3 then the constant ⅔ in (1.2) can be replaced by 1/2.2. For Q1 equality is needed on at least one side of (1.2) while for Q2, Q3 the constant ⅔ can be reduced to 12/25 but no further.

The positive values of inhomogeneous ternary quadratic forms

Let f(x, y, z) be an indefinite ternary quadratic form of signature (2, 1) and determinant d ≠ 0. Davenport [3] has shown that there exist integral x, y, z with, the equality sign being necessary if and only if f is a positive multiple of f1(x, y, z) = x2 + yz.