If $\mu$, $\nu$, $\sigma$ denote \emph{different} suffixes we get the following possible cases (the summation convention being suspended): %[** TN: Punctuation moved in-line] \[ \left. \begin{aligned} \{\mu\mu, \mu\} &= \Neg\tfrac{1}{2} g^{\mu\mu}\, \frac{\dd g_{\mu\mu}}{\dd x_{\mu}} = \tfrac{1}{2} \frac{\dd}{\dd x_{\mu}} (\log g_{\mu\mu}), \\ \{\mu\mu, \nu\} &= -\tfrac{1}{2} g^{\mu\nu}\, \frac{\dd g_{\mu\mu}}{\dd x_{\nu}}, \\ \{\mu\nu, \nu\} &= \Neg\tfrac{1}{2} g^{\nu\nu}\, \frac{\dd g_{\nu\nu}}{\dd x_{\mu}} = \tfrac{1}{2} \frac{\dd}{\dd x_{\mu}} (\log g_{\nu\nu}), \\ \{\mu\nu, \sigma\} &= \Neg0. \end{aligned}\right\} \Tag{(38.4)} \] It is now easy to go systematically through the $40$ $3$-index symbols calculating the values of those which do not vanish. We obtain the following results, the accent denoting differentiation with respect to~$r$: %[** TN: Punctuation moved in-line] \[ \left. \begin{aligned} \{11, 1\} &= \tfrac{1}{2} \lambda', \\ \{12, 2\} &= 1/r, \\ \{13, 3\} &= 1/r, \\ \{14, 4\} &= \tfrac{1}{2} \nu', \\ \{22, 1\} &= -re^{-\lambda}, \\ \{23, 3\} &= \cot\theta, \\ \{33, 1\} &= -r\sin^{2}\theta\, e^{-\lambda}, \\ \{33, 2\} &= -\sin\theta \cos\theta, \\ \{44, 1\} &= \tfrac{1}{2} e^{\nu-\lambda} \nu'. \end{aligned} \right\} \Tag{(38.5)} \] The remaining $31$~symbols vanish. Note that $\{21, 2\}$ is the same as $\{12, 2\}$, etc. These values must be substituted in~\Eq{(37.2)}. As there may be some pitfalls in carrying this out, we shall first write out the equations~\Eq{(37.2)} in full, omitting the terms ($223$~in number) which now obviously vanish. %[** TN: Re-broken] \begin{align*} G_{11} &= -\frac{\dd}{\dd r} \{11, 1\} \begin{aligned}[t] &+ \{11, 1\} \{11, 1\} + \{12, 2\} \{12, 2\} \\ &+ \{13, 3\} \{13, 3\} + \{14, 4\} \{14, 4\} \end{aligned} \\ &\qquad + \frac{\dd^{2}}{\dd r^{2}} \log\sqrt{-g} - \{11, 1\}\, \frac{\dd}{\dd r} \log\sqrt{-g},\displaybreak[0] \\ G_{22} &= -\frac{\dd}{\dd r} \{22, 1\} + 2\{22, 1\} \{21, 2\} + \{23, 3\} \{23, 3\} \\ &\qquad + \frac{\dd^{2}}{\dd\theta^{2}} \log\sqrt{-g} - \{22, 1\}\, \frac{\dd}{\dd r} \log\sqrt{-g},\displaybreak[0] \\ G_{33} &= -\frac{\dd}{\dd r} \{33, 1\} - \frac{\dd}{\dd\theta}\, \{33, 2\} \\ &\qquad + 2\{33, 1\} \{31, 3\} + 2\{33, 2\} \{32, 3\} \\ &\qquad - \{33, 1\}\, \frac{\dd}{\dd r} \log\sqrt{-g} - \{33, 2\}\, \frac{\dd}{\dd\theta} \log\sqrt{-g},\displaybreak[0] \\ G_{44} &= -\frac{\dd}{\dd r} \{44, 1\} + 2\{44, 1\} \{41, 4\} - \{44, 1\}\, \frac{\dd}{\dd r} \log\sqrt{-g},\displaybreak[0] \\ G_{12} &= \Neg \{13, 3\} \{23, 3\} - \{12, 2\}\, \frac{\dd}{\dd\theta} \log\sqrt{-g}. \end{align*} The remaining components contain no surviving terms. \PageSep{85}