{\Loosen The contracted Riemann-Christoffel tensor is formed by setting $\epsilon = \sigma$ in~$B_{\mu\nu\sigma}^{\epsilon}$. It is denoted by~$G_{\mu\nu}$. Hence by~\Eq{(34.4)}} \[ G_{\mu\nu} = \{\mu\sigma, \alpha\} \{\alpha\nu, \sigma\} - \{\mu\nu, \alpha\} \{\alpha\sigma, \sigma\} + \frac{\dd}{\dd x_{\nu}} \{\mu\sigma, \sigma\} - \frac{\dd}{\dd x_{\sigma}} \{\mu\nu, \sigma\}. \Tag{(37.1)} \] The symbols containing a duplicated suffix are simplified by~\Eq{(35.4)}, viz.\ \[ \{\mu\sigma, \sigma\} = \frac{\dd}{\dd x_{\mu}} \log \sqrt{-g}. \] Hence, with some alterations of dummy suffixes, %[** TN: Not broken in the original] \begin{multline*} G_{\mu\nu} = -\frac{\dd}{\dd x_{\alpha}} \{\mu\nu, \alpha\} + \{\mu\alpha, \beta\} \{\nu\beta, \alpha\} \\ + \frac{\dd^{2}}{\dd x_{\mu}\, \dd x_{\nu}} \log \sqrt{-g} - \{\mu\nu, \alpha\}\, \frac{\dd}{\dd x_{\alpha}} \log \sqrt{-g}. \Tag{(37.2)} \end{multline*} Contraction by setting $\epsilon = \mu$ does not provide an alternative tensor, because \[ B_{\mu\nu\sigma}^{\mu} = g^{\mu\rho} B_{\mu\nu\sigma\rho} = 0, \] owing to the antisymmetry of~$B_{\mu\nu\sigma\rho}$ in $\mu$ and~$\rho$. The law \[ G_{\mu\nu} = 0, \Tag{(37.3)} \] in empty space, is chosen by Einstein for his law of gravitation. We see from~\Eq{(37.2)} that $G_{\mu\nu}$~is a symmetrical tensor; consequently the law provides $10$~partial differential equations to determine the~$g_{\mu\nu}$. It will be found later (\SecRef{52}) that there are $4$~identical relations between them, so that the number of equations is effectively reduced to~$6$. The equations are of the second order and involve the second differential coefficients of~$g_{\mu\nu}$ linearly. We proved in \SecRef{36} that tensors not containing derivatives beyond the second must necessarily be compounded from $g_{\mu\nu}$ and~$B_{\mu\nu\sigma}^{\epsilon}$; so that, unless we are prepared \PageSep{82} to go beyond the second order, the choice of a law of gravitation is very limited, and we can scarcely avoid relying on the tensor~$G_{\mu\nu}$\footnotemark.\footnotetext {The law $B_{\mu\nu\sigma\rho} = 0$ (giving flat space-time throughout all empty regions) would obviously be too stringent, since it does not admit of the existence of irreducible fields of force.} Without introducing higher derivatives, which would seem out of place in this problem, we can suggest as an alternative to~\Eq{(37.3)} the law \[ G_{\mu\nu} = \lambda g_{\mu\nu}, \Tag{(37.4)} \] where $\lambda$~is a universal constant. There are theoretical grounds for believing that this is actually the correct form; but it is certain that $\lambda$~must be an extremely small constant, so that in practical applications we still take \Eq{(37.3)} as sufficiently approximate. The introduction of the small constant~$\lambda$ leads to the spherical world of Einstein or de~Sitter to which we shall return in Chapter~\ChapNum{V}\@.