the planet and $a$ is its semi–major axis. Note that the function $ \left( \sin^2 I_0 - 2/5
\right)^{-1/2}$ decreases very sharply from infinity to $\sim 3$ as $I_0$ increases from $I_c$ to about 45$^{\circ}$ and then decreases by about 50% as $I_0$ continues to increase up to 90$^{\circ}$.

The $Z$–component of the angular momentum of the orbit, $L_z \propto
\sqrt{1-e^2} \cos I$, where $I$ is the inclination angle between the orbit and the plane of the disc, is constant. Therefore $I$ also oscillates with time and is out of phase with $e$.

Numerical simulations {#sec:simulations}
=====================

We consider a star of mass $M_{\star}=1$ M$_{\sun}$ surrounded by a disc of mass $M_{\rm disc}$ and a planet of mass $M_p$ whose orbit is inclined with respect to the disc. The planet interacts gravitationally with the star and the disc but we take $M_p \ll M_{\rm disc}$ so that it has no effect on the disc. To study the evolution of the system, we use the $N$–body code described in Papaloizou & Terquem (2001) in which we have added the gravitational force exerted by the disc onto the planet.

The equation of motion for the planet is: $${d^2 {\bf r} \over dt^2} = -{GM_\star{\bf r} \over |{\bf r}|^3} -
\mbox{\boldmath $\nabla$} \Phi - {GM_p{\bf r} \over |{\bf r}|^3} + {\bf
\Gamma}_{t,r} \; , \label{emot}$$

where ${\bf r}$ is the position vector of the planet and $\Phi$ is the gravitational potential of the disc given by equation (\[Phi\]) with $R_i$ and $R_o$ being the inner and outer radii of the disc. The third term on the right–hand side is the acceleration of the coordinate system based on the central star. Tides raised by the star in the planet and relativistic effects are included through ${\bf \Gamma}_{t,r}$, but they are unimportant here as the planet does not approach the star closely. Equation (\[emot\]) is integrated using the Bulirsch–Stoer method and the integrals involved in $\mbox{\boldmath $\nabla$} \Phi$ are calculated with the Romberg method (Press et al. 1993). In most runs, the 