= \frac{1}{n_0 K_p} \ln\left(\frac{n_{0}+n_{i}}{n_0 - n_i}\right),
\label{eq:trgh-p}$$ by just setting $\overline{n}=0$ in Eq. (\[eq:gh-solution-p\]).

Similarly, in the case of $n_i<0$, only the rolling develops and the pitching is irrelevant, thus we obtain $\overline{n}(t)$ and the time for reversal $t_{rGH-}$ as $$\overline{n}(t) = -n_{0}\frac{(n_{0} + |n_{i}|)\exp(-n_{0}K_{r}t) - (n_{0} - |n_{i}|) }{(n_{0} + |n_{i}|)\exp(-n_{0}K_{r}t) + (n_{0} - |n_{i}|)} 
\label{eq:gh-solution-m}$$ and $$t_{rGH-} =\frac{1}{n_0 K_r} \ln\left(\frac{n_0+|n_i|}{n_0 - |n_i|} \right). \label{eq:trgh-m}$$ Equations (\[eq:trgh-p\]) and (\[eq:trgh-m\]) are Garcia-Hubbard formulas for the times for reversal [@GarciaHubbard1988].

From the expressions of $K_{p}$ and $K_{r}$ given by Eqs. (\[eq:Kp\]) and (\[eq:Kr\]), we immediately notice that (1) the time for reversal is inversely proportional to the skew angle $\xi$ in the small skewness regime, and (2) the ratio of the time for reversal $t_{rGH-}/t_{rGH+}$ is simply given by the squared ratio of the pitch frequency to the roll frequency $\omega_{p}^2 / \omega_{r}^2$, provided initial values $n_{0}$ and $n_{i}$ are the same except their signs.

For a typical rattleback, $\omega_{p}^2 \gg \omega_{r}^2$, thus $t_{rGH+} \ll t_{rGH-}$, i.e., the time for reversal is much shorter in the case of $n_{i}>0$ than in the case of $n_{i}<0$. Thus we call the spin direction of $n_{i} > 0$ the *unsteady direction* [@GarciaHubbard1988], and that 