We introduce two expressions (not tensors) of great importance throughout our subsequent work, namely \begin{align*} [\mu\nu, \sigma] &= \tfrac{1}{2} \left(\frac{\dd g_{\mu\sigma}}{\dd x_{\nu}} + \frac{\dd g_{\nu\sigma}}{\dd x_{\mu}} - \frac{\dd g_{\mu\nu}}{\dd x_{\sigma}}\right), \Tag{(27.1)} \\ \{\mu\nu, \sigma\} &= \tfrac{1}{2} g^{\sigma\lambda} \left(\frac{\dd g_{\mu\lambda}}{\dd x_{\nu}} + \frac{\dd g_{\nu\lambda}}{\dd x_{\mu}} - \frac{\dd g_{\mu\nu}}{\dd x_{\lambda}}\right). \Tag{(27.2)} \end{align*} \PageSep{59} We have \begin{align*} \{\mu\nu, \sigma\} &= g^{\sigma\lambda}\, [\mu\nu, \lambda], \Tag{(27.3)} \\ [\mu\nu, \sigma] &= g_{\sigma\lambda}\, \{\mu\nu, \lambda\}. \Tag{(27.4)} \end{align*} The result~\Eq{(27.3)} is obvious from the definitions. To prove~\Eq{(27.4)}, multiply \Eq{(27.3)} by~$g_{\sigma\alpha}$; then \begin{align*} g_{\sigma\alpha}\, \{\mu\nu, \sigma\} &= g_{\sigma\alpha} g^{\sigma\lambda}\, [\mu\nu, \lambda] \\ &= g_{\alpha}^{\lambda}\, [\mu\nu, \lambda] \\ &= [\mu\nu, \alpha], \end{align*} which is equivalent to~\Eq{(27.4)}. Comparing with \Eq{(26.1)} and~\Eq{(26.2)} we see that the passage from the ``square'' to the ``curly'' symbol, and \Foreign{vice versa}, is the same process as raising and lowering a suffix. It might be convenient to use a notation in which this was made evident, e.g.\ \[ \Gamma_{\mu\nu, \sigma} = [\mu\nu, \sigma],\quad \Gamma_{\mu\nu}^{\sigma} = \{\mu\nu, \sigma\}, \] but we shall adhere to the more usual notation. From \Eq{(27.1)} it is found that \[ [\mu\nu, \sigma] + [\sigma\nu, \mu] = \frac{\dd g_{\mu\sigma}}{\dd x_{\nu}}. \Tag{(27.5)} \] There are $40$~different $3$-index symbols of each kind. It may here be explained that the~$g_{\mu\nu}$ are components of a generalised \emph{potential}, and the \index{Potential!gravitational}% $3$-index symbols components of a generalised \emph{force} in the gravitational theory (see \SecRef{55}). \Section{28.}{Equations of a geodesic} We shall now determine the equations of a geodesic or path between two points for which \[ \int ds \text{ is stationary.} \] This absolute track is of fundamental importance in dynamics, but at the moment we are concerned with it only as an aid in the development of the tensor calculus\footnotemark.\footnotetext {Our ultimate goal is equation~\Eq{(29.3)}. An alternative proof (which does not introduce the calculus of variations) is given in \SecRef{31}.}