Let $K$ be a totally real algebraic number field whose ring of integers $R$ is a principal ideal domain. Let $f(x_1,x_2,x_3)$ be a totally definite ternary quadratic form with coefficients in $R$. We shall study representations of totally positive elements $N\in R$ by $f$. We prove a quantitative formula relating the number of representations of $N$ by different classes in the genus of $f$ to the class number of $R\left[\sqrt{-c_{f}N}\right]$, where $c_f\in R$ is a constant depending only on $f$. We give an algebraic proof of a classical result of H. Maass on representations...