(2) For the first term of the action, $\Star{G}^{2} \sqrt{-g}$~was chosen instead of the simpler $\Star{G} \sqrt{-g}$, \emph{because the latter is not an in-invariant-density} and cannot be regarded as a measure of any absolute property of the region. It is interesting to trace how this improvement leads to the appearance of the term $\delta (-2\lambda \sqrt{-g})$ in~\Eq{(90.3)}, so that the cosmical curvature-term in the expression for the energy-tensor now appears quite naturally and inevitably. We may contrast this with the variation of~$G \sqrt{-g}$ worked out in \SecRef{60}, where no such term appears. In attributing more fundamental importance to the in-invariant $\Star{G}^{2} \sqrt{-g}$ than to the co-invariant $\Star{G} \sqrt{-g}$, Weyl's theory makes an undoubted advance towards the truth. \PageSep{213} \Part{II.}{Generalised Theory} \Section{91.}{Parallel displacement} \index{Displacement!parallel}% \index{Equivalence of displacements}% \index{Generalisation of Weyl's theory}% \index{Parallel displacement}% Let an infinitesimal displacement~$A^{\mu}$ at the point~$P$ (coordinates,~$x_{\mu}$) be carried by parallel displacement to a point~$P'$ (coordinates, $x_{\mu} + dx_{\mu}$) infinitely near to~$P$. The most general possible continuous formula for the change of~$A^{\mu}$ is of the form \[ dA^{\mu} = -\Gamma_{\nu\alpha}^{\mu} A^{\alpha}\, dx_{\nu}, \Tag{(91.1)} \] where $\Gamma_{\nu\alpha}^{\mu}$, which is not assumed to be a tensor, represents $64$~arbitrary coefficients. Both $A^{\alpha}$ and~$dx_{\nu}$ are infinitesimals, so that there is no need to insert any terms of higher order. We are going to build the theory afresh starting from this notion of infinitesimal parallel displacement; and by so doing we arrive at a generalisation even wider than that of Weyl. Our fundamental axiom is that parallel displacement has some significance in regard to the ultimate structure of the world---it does not much matter what significance. The idea is that out of the whole group of displacements radiating from~$P'$, we can select one $A^{\mu} + dA^{\mu}$ which has some kind of \emph{equivalence} to the displacement~$A^{\mu}$ at~$P$. We do not define the nature of this equivalence, except that it shall have reference to the part played by~$A^{\mu}$ in the relation-structure which underlies the world of physics. Notice that--- (1) This equivalence is only supposed to exist in the limit when $P$ and~$P'$ are infinitely near together. For more distant points equivalence can in general only be approximate, and gradually becomes indeterminate as the distance is increased. It can be made determinate by specifying a particular route of connection, in which case the equivalence is traced step by step along the route. (2) The equivalence is not supposed to exist between any world-relations other than displacements. Hitherto we have applied parallel displacement to any tensor, but in this theory we only use it for displacements.