Weyl adopts an action-density \[ A \sqrt{-g} = (\Star{G}^{2} - \alpha F_{\mu\nu} F^{\mu\nu}) \sqrt{-g}, \Tag{(90.1)} \] the constant~$\alpha$ being a pure number. He makes the hypothesis that it obeys \PageSep{210} the principle of stationary action for all variations $\delta g_{\mu\nu}$, $\delta\kappa_{\mu}$ which vanish at the boundary of the region considered. Accordingly \[ \frac{\Ham A}{\Ham g_{\mu\nu}} = 0,\quad \frac{\Ham A}{\Ham \kappa_{\mu}} = 0. \Tag{(90.2)} \] Weyl himself states that his action-principle is probably not realised in nature exactly in this form. But the procedure is instructive as showing the kind of unifying principle which is aimed at according to one school of thought. The variation of $\Star{G}^{2} \sqrt{-g}$ is \[ 2\, \Star{G}\, \delta(\Star{G} \sqrt{-g}) - \Star{G}^{2}\, \delta(\sqrt{-g}), \] which in the natural gauge becomes by~\Eq{(89.1)} \[ 8\lambda\, \delta(\Star{G} \sqrt{-g}) - 16\lambda^{2}\, \delta(\sqrt{-g}). \] Hence by~\Eq{(87.6)} \[ \frac{1}{8\lambda}\, \delta(A \sqrt{-g}) = \delta \bigl\{(G - 6\kappa_{\alpha}^{\alpha} + 6\kappa_{\alpha} \kappa^{\alpha} - 2\lambda - \beta F_{\mu\nu} F^{\mu\nu}) \sqrt{-g}\bigr\}, \Tag{(90.3)} \] where $\beta = \alpha/8\lambda$. The term $\kappa_{\alpha}^{\alpha} \sqrt{-g}$ can be dropped, because by~\Eq{(51.11)} \[ \kappa_{\alpha}^{\alpha} \sqrt{-g} = \frac{\dd}{\dd x_{\alpha}} (\kappa^{\alpha} \sqrt{-g}). \] This can be integrated, and yields a surface-integral over the boundary of the region considered. Its Hamiltonian derivatives accordingly vanish.