pitch frequency is always higher for an ellipsoid with a uniform mass density whose surface is given by $x^2/c^2
+ y^2/b^2+ z^2/a^2 = 1\ (b^2 > c^2 > a^2)$. This also holds for a semi-ellipsoid for $b^2 > c^2 > (5/8)a^2$, where the co-ordinate system is the same as the ellipsoid.

### Time for reversal

Now we study the time evolution of the *spin* $n$ defined as the vertical component of the angular velocity $$n \equiv \bm{u}\cdot\bm{\omega},$$ assuming that the expressions for the asymmetric torque coefficients, $K_p$ and $K_r$, obtained above are valid even when $\omega_{z}\ne 0$. We consider the quantities $\overline{n}$, $\overline{E}_p$, and $\overline{E}_r$, averaged over the time scale much longer than the oscillation periods, yet much shorter than the time scale for spin change. Then, these averaged quantities should follow the following evolution equations: $$\begin{aligned}
 I_{\textrm{eff}} \frac{d\overline{n}(t)}{dt} &= - K_{p} \overline{E}_{p}(t) + K_{r} \overline{E}_{r}(t) , \label{eq:diff-ne-1}\\
 \frac{d \overline{E}_{p}(t)}{dt} & =  K_{p} \overline{n}(t) \overline{E}_{p}(t),\label{eq:diff-ne-2}\\
 \frac{d \overline{E}_{r}(t)}{dt} & = - K_{r} \overline{n}(t) \overline{E}_{r}(t) \label{eq:diff-ne-3}.\end{aligned}$$ Here, $I_{\textrm{eff}}$ is the effective moment of inertia around $\bm{u}$ under the existence of the oscillations, and is assumed to be constant; it should be close to $I_{zz}$. As can be seen easily, the total energy 