Here we have used \[ \frac{\dd g_{\alpha\beta}}{\dd x_{\lambda}'} = \frac{\dd g_{\alpha\beta}}{\dd x_{\gamma}}\, \frac{\dd x_{\gamma}}{\dd x_{\lambda}'}, \] and further we have interchanged the dummy suffixes $\alpha$ and~$\beta$ in the second term in the bracket. Similarly \begin{align*} \frac{\dd g_{\nu\lambda}'}{\dd x_{\mu}'} &= g_{\alpha\beta} \left\{ \frac{\dd^{2} x_{\alpha}}{\dd x_{\mu}'\, \dd x_{\nu}'}\, \frac{\dd x_{\beta}}{\dd x_{\lambda}'} + \frac{\dd^{2} x_{\alpha}}{\dd x_{\mu}'\, \dd x_{\lambda}'}\, \frac{\dd x_{\beta}}{\dd x_{\nu}'}\right\} + \frac{\dd x_{\alpha}}{\dd x_{\mu}'}\, \frac{\dd x_{\beta}}{\dd x_{\nu}'}\, \frac{\dd x_{\gamma}}{\dd x_{\lambda}'}\, \frac{\dd g_{\beta\gamma}}{\dd x_{\alpha}}, \Tag{(31.12)} \\ \frac{\dd g_{\nu\lambda}'}{\dd x_{\nu}'} &= g_{\alpha\beta} \left\{ \frac{\dd^{2} x_{\alpha}}{\dd x_{\nu}'\, \dd x_{\mu}'}\, \frac{\dd x_{\beta}}{\dd x_{\lambda}'} + \frac{\dd^{2} x_{\alpha}}{\dd x_{\nu}'\, \dd x_{\lambda}'}\, \frac{\dd x_{\beta}}{\dd x_{\mu}'}\right\} + \frac{\dd x_{\alpha}}{\dd x_{\mu}'}\, \frac{\dd x_{\beta}}{\dd x_{\nu}'}\, \frac{\dd x_{\gamma}}{\dd x_{\lambda}'}\, \frac{\dd g_{\alpha\gamma}}{\dd x_{\beta}}. \Tag{(31.13)} \end{align*} % [** TN: [sic] Add and subtract] Add \Eq{(31.12)} and \Eq{(31.13)} and subtract~\Eq{(31.11)}, we obtain by~\Eq{(27.1)} \[ [\mu\nu, \lambda]' = g_{\alpha\beta}\, \frac{\dd x_{\alpha}}{\dd x_{\mu}'\, \dd x_{\nu}'}\, \frac{\dd x_{\beta}}{\dd x_{\lambda}'} + \frac{\dd x_{\alpha}}{\dd x_{\mu}'}\, \frac{\dd x_{\beta}}{\dd x_{\nu}'}\, \frac{\dd x_{\gamma}}{\dd x_{\lambda}'}\, [\alpha\beta, \gamma]. \Tag{(31.2)} \] \PageSep{66} %[** TN: [sic] Multiply] Multiply through by $g'^{\lambda\rho}\, \dfrac{\dd x_{\epsilon}}{\dd x_{\rho}'}$, we have by~\Eq{(27.3)} %[** TN: Added break in first line] \begin{align*} \{\mu\nu, \rho\}\, \frac{\dd x_{\epsilon}}{\dd x_{\rho}'} &= g_{\alpha\beta}\, \frac{\dd^{2} x_{\alpha}}{\dd x_{\mu}'\, \dd x_{\nu}'} · g'^{\lambda\rho}\, \frac{\dd x_{\beta}}{\dd x_{\lambda}'}\, \frac{\dd x_{\epsilon}}{\dd x_{\rho}'} \\ &\qquad\qquad+ g'^{\lambda\rho}\, \frac{\dd x_{\gamma}}{\dd x_{\lambda}'}\, \frac{\dd x_{\epsilon}}{\dd x_{\rho}'} · \frac{\dd x_{\alpha}}{\dd x_{\mu}'}\, \frac{\dd x_{\beta}}{\dd x_{\nu}'}\, [\alpha\beta, \gamma]\displaybreak[0] \\ &= g_{\alpha\beta} g^{\beta\epsilon}\, \frac{\dd^{2} x_{\alpha}}{\dd x_{\mu}'\, \dd x_{\nu}'} + \frac{\dd x_{\alpha}}{\dd x_{\mu}'}\, \frac{\dd x_{\beta}}{\dd x_{\nu}'}\, g^{\gamma\epsilon}\, [\alpha\beta, \gamma] \qquad\text{by \Eq{(23.21)}}\displaybreak[0] \\ &= \frac{\dd^{2} x_{\epsilon}}{\dd x_{\mu}'\, \dd x_{\nu}'} + \frac{\dd x_{\alpha}}{\dd x_{\mu}'}\, \frac{\dd x_{\beta}}{\dd x_{\nu}'}\, \{\alpha\beta, \epsilon\}, \Tag{(31.3)} \end{align*} a formula which determines the second derivative $\dd^{2} x_{\epsilon}/\dd x_{\mu'}'\, \dd x_{\nu}'$ in terms of the first derivatives.