We have to bear in mind the two aspects of action in this subject. It is primarily a physical quantity having a definite numerical value, given indifferently by \Eq{(60.11)} or~\Eq{(60.2)}, which is of special importance because it is invariant. But it also denotes a mathematical function of the variables; the functional form, which is all important, will differ according to which of the two expressions is used. In particular we have to consider the partial derivatives, and these will depend on the variables in terms of which the action is expressed. The Hamiltonian method of variation of an integral is of great importance in this subject; several examples of it will be given presently. I think it is unfortunate that this valuable method is nearly always applied in the form of a principle of stationary action. By considering the variation of the integral for small variations of the~$g_{\mu\nu}$, or other variables, we obtain a kind of generalised differential coefficient which I will call the Hamiltonian derivative. It may be possible to construct integrals for which the Hamiltonian derivatives vanish, so that the integral has the stationary property. But just as in the ordinary differential calculus we are not solely concerned with problems of maxima and minima, and we take some interest in differential coefficients which do not vanish; so Hamiltonian derivatives may be worthy of attention even when they disappoint us by failing to vanish. Let us consider the variation of the gravitational action in a region, viz.\ \[ 8\pi\, \delta A = \delta \int G\sqrt{-g}\, d\tau, \] for arbitrary small variations~$\delta g_{\mu\nu}$ which vanish at and near\footnote {So that their first derivatives also vanish.} the boundary of the region. By~\Eq{(58.8)} \[ \delta \int G\sqrt{-g}\, d\tau = -\delta \int \mf{L}\, d\tau + \delta \int \frac{\dd}{\dd x_{\alpha}} \left(\mf{g}^{\mu\nu}\, \frac{\dd\mf{L}}{\dd\mf{g}_{\alpha}^{\mu\nu}}\right) d\tau. \] Also since $\mf{L}$~is a function of $\mf{g}^{\mu\nu}$ and~$\mf{g}_{\alpha}^{\mu\nu}$ \[ \int \delta\mf{L}\, d\tau = \int \left( \frac{\dd\mf{L}}{\dd\mf{g}^{\mu\nu}}\, \delta\mf{g}^{\mu\nu} + \frac{\dd\mf{L}}{\dd\mf{g}_{\alpha}^{\mu\nu}}\, \delta\mf{g}_{\alpha}^{\mu\nu} \right) d\tau, \] and, by partial integration of the second term, \[ = \int \left( \frac{\dd\mf{L}}{\dd\mf{g}^{\mu\nu}} - \frac{\dd}{\dd x_{\alpha}}\, \frac{\dd\mf{L}}{\dd\mf{g}_{\alpha}^{\mu\nu}}\right) \delta\mf{g}^{\mu\nu}\, d\tau + \int \frac{\dd}{\dd x_{\alpha}} \left(\frac{\dd\mf{L}}{\dd\mf{g}_{\alpha}^{\mu\nu}}\, \delta\mf{g}^{\mu\nu}\right) d\tau. \] \PageSep{139} By~\Eq{(58.6)} the first integrand becomes $-G_{\mu\nu}\, \delta\mf{g}^{\mu\nu}$, so that we have \[ \delta \int G\sqrt{-g}\, d\tau = \int G_{\mu\nu}\, \delta(g^{\mu\nu} \sqrt{-g})\, d\tau + \int \frac{\dd}{\dd x_{\alpha}} \left(\mf{g}^{\mu\nu}\, \delta\left(\frac{\dd\mf{L}}{\dd\mf{g}_{\alpha}^{\mu\nu}}\right)\right) d\tau. \Tag{(60.3)} \]