  In this paper, the spectrum of the following fourth order problem
\begin{equation*}
  \begin{cases}
  \Delta^2 u+\nu u=-\lambda \Delta u &\text{in } D_1,\newline
  u=\partial_r u= 0 &\text{on } \partial D_1,
  \end{cases} \end{equation*} where $D_1$ is the unit ball in ${\mathbb R}^N$,
is determined for $\nu < 0$ as well as the nodal properties of the
corresponding eigenfunctions. In particular, we show that the first eigenvalue
is simple and that the corresponding eigenfunction is radial and (up to a
multiplicative factor) positive and decreasing with respect to the radius. This
completes earlier results obtained for $\nu \ge 0$ and for $\nu <0$.
