The second term can be integrated immediately giving a triple integral over the boundary of the four-dimensional region; and it vanishes because all variations vanish at the boundary by hypothesis. Hence \begin{align*} \delta \int G\sqrt{-g}\, d\tau &= \Neg \int G_{\mu\nu}\, \delta(g^{\mu\nu} \sqrt{-g})\, d\tau \Tag{(60.41)} \\ &= -\int (G_{\mu\nu} - \tfrac{1}{2}g^{\mu\nu}G)\, \delta g_{\mu\nu} \sqrt{-g}\, d\tau \Tag{(60.42)} \end{align*} by~\Eq{(58.91)}. I call the coefficient $-(G^{\mu\nu} - \frac{1}{2}g^{\mu\nu}G)$ the \emph{Hamiltonian derivative} of~$G$ with \index{H@$\Ham$ (Hamiltonian operator)}% \index{Hamiltonian derivative}% \index{Operators!H@$\Ham$}% respect to~$g_{\mu\nu}$, writing it symbolically \[ \frac{\Ham G}{\Ham g_{\mu\nu}} = -(G^{\mu\nu} - \tfrac{1}{2}g^{\mu\nu}G) = 8\pi T^{\mu\nu}. \Tag{(60.43)} \] We see from~\Eq{(60.42)} that the action~$A$ is only stationary when the energy-tensor~$T^{\mu\nu}$ \index{Action, principle of Stationary}% \index{Stationary action, principle of}% vanishes, that is to say in empty space. In fact action is only stationary when it does not exist---and not always then. It would thus appear that the Principle of Stationary Action is in general \index{Principle!of least action}% untrue. Nevertheless some modified statement of the principle appears to have considerable significance. In the actual world the space occupied by matter (electrons) is extremely small compared with the empty regions. Thus the Principle of Stationary Action, although not universally true, expresses a very general tendency---a tendency with exceptions\footnotemark.\footnotetext {I do not regard electromagnetic fields as constituting an exception, because they have not yet been taken into account in our work. But the action of matter has been fully included, so that the break-down of the principle as applied to matter is a definite exception.} Our theory does not account for this atomicity of matter; and in the stationary variation of action \index{Atomicity}% we seem to have an indication of a way of approaching this difficult problem, although the precise formulation of the law of atomicity is not yet achieved. It is suspected that it may involve an ``action'' which is capable only of discontinuous variation.