to @halphen_traite_1886 [chapter VII] (and explained in detail in @greenhill_applications_1959 [chapter VII]), involves the decomposition of $A$ and $B$ into separate fractions, resulting in the split of the integral into fundamental forms that can be integrated using the Weierstrassian functions.

It is again easier to use the simplified solutions and , and thus obtain the time equation $$\begin{aligned}
t &= \int_0^\tau\left[\xi^{2}\left(u\right)+\eta^{2}\left(u\right)\right]du \\
& = \left(\xi_r^2+\eta_r^2\right)\tau + \frac{1}{2}\int_0^\tau
\frac{f_{\xi}^{\prime}\left(\frac{\xi_{r}^{2}}{2}\right)}{\wp_{\xi}\left(u-\tau_{\xi}\right)
-\frac{1}{24}f_{\xi}^{\prime\prime}\left(\frac{\xi_{r}^{2}}{2}\right)}du\notag\\
&\quad +\frac{1}{2}\int_0^\tau
\frac{f_{\eta}^{\prime}\left(\frac{\eta_{r}^{2}}{2}\right)}{\wp_{\eta}\left(u-\tau_{\eta}\right)
-\frac{1}{24}f_{\eta}^{\prime\prime}\left(\frac{\eta_{r}^{2}}{2}\right)}du.\label{eq:tau_01}\end{aligned}$$ The integrals appearing in eq. are known and they can be computed directly. To this end, it would be tempting to apply the formulæ in @gradshtein_table_2007 [§5.141]. However, as it can be verified by direct substitution using the exact solution of the cubic equation[^7], $\frac{1}{24}f_{\xi}^{\prime\prime}\left(\frac{\xi_{r}^{2}}{2}\right)$ and $\frac{1}{24}f_{\eta}^{\prime\prime}\left(\frac{\eta_{r}^{2}}{2}\right)$ are always roots of the characteristic cubic equations $$4t^3 - g_2t - g_3 = 0,\label{eq:char_cubic}$$ associated to $\wp_\xi$ and $\wp_\eta$. Consequently, the formulæ in @gradshtein_table_2007 [§5.141] will be singular, and we have to use instead the results in @jules_tannery_elements_1893 [§CXII], which yield the formula $$\int\frac{du}{\wp\left(u\right)-e_i} = \frac{1}{g_2/4-3e_i^2}\left[ue_i + \zeta\left(u-\omega_i\right)\right].$$ In this formula, the $e_i$ represents the three roots of the characteristic cubic equation, while the $\omega_i$ are defined by the relation $e_i=\wp\left(\omega_i\right)$ (so that, following @abramowitz_handbook_1964 [eq. 18.3.1], two of 