We have thus demonstrated the general theorem--- \emph{The Hamiltonian derivative of any fundamental invariant is a tensor whose \index{Hamiltonian derivative!of fundamental invariants}% divergence vanishes.} \index{Divergence!Hamiltonian@of Hamiltonian derivative of an invariant}% The theorem of \SecRef{52} is a particular case, since $T^{\mu\nu}$~is the Hamiltonian derivative of~$G$ by~\Eq{(60.43)}. \Section{62.}{Alternative energy-tensors} \index{Energy-tensor of matter}% \index{Fundamental velocity!invariants}% We have hitherto identified the energy-tensor with $G_{\mu}^{\nu} - \tfrac{1}{2} g_{\mu}^{\nu}G$ mainly because the divergence of the latter vanishes identically; but the theorem just proved enables us to derive other fundamental tensors whose divergence vanishes, so that alternative identifications of the energy-tensor would seem to be possible. The three simplest fundamental invariants are \[ K = G,\quad K' = G_{\mu\nu} G^{\mu\nu},\quad K'' = B_{\mu\nu\sigma}^{\rho} B_{\rho}^{\mu\nu\sigma}. \Tag{(62.1)} \] Hitherto we have taken $\Ham K/\Ham g_{\mu\nu}$ to be the energy-tensor; but if $\Ham K'/\Ham g_{\mu\nu}$ were substituted, the laws of conservation of energy and momentum would be satisfied, since the divergence vanishes. Similarly $\Ham K''/\Ham g_{\mu\nu}$ could be used. The condition for empty space is given by the vanishing of the energy-tensor. Hence for the three possible hypotheses, the law of gravitation in empty space is \[ \frac{\Ham K}{\Ham g_{\mu\nu}},\quad \frac{\Ham K'}{\Ham g_{\mu\nu}},\quad \frac{\Ham K''}{\Ham g_{\mu\nu}} = 0 \Tag{(62.2)} \] respectively. It is easy to see that the last two tensors contain fourth derivatives of the~$g_{\mu\nu}$; so that if we can lay it down as an essential condition that the law of gravitation in empty space must be expressed by differential equations of the \PageSep{142} second order, the only possible energy-tensor is the one hitherto accepted. For fourth-order equations the question of the nature of the boundary conditions necessary to supplement the differential equations would become very difficult; but this does not seem to be a conclusive reason for rejecting such equations.