Let a family of polynomials be (1) \(P(s)=t_ 0s^ n+t_ 1s^{n- 1}+...+t_ n\) where \(0<\alpha_ j\leq t_ j\leq \beta_ j\). Recently, the authors [ibid. 33, No.5, 509-511 (1988; Zbl 0643.93059)] have shown that a necessary and sufficient condition for (1) to have a damping ratio of \(\phi\) is that the \(2^{n+1}\) polynomials in (1) which have \(t_ k=\alpha_ k\) or \(t_ k=\beta_ k\) have a damping ratio of \(\phi\). This note derives a more powerful result requiring only eight polynomials to be Hurwitz for (1) to have a damping ratio of \(\phi\) using \textit{V. L. Kharitonov}'s [Izv. Akad. Nauk Kaz. SSR, Ser. Fiz.-Mat. 1978, No.1, 53-57 (1978; Zbl 0388.30005)] theorem for complex polynomials.