(3) It is not assumed that there is any complete observational test of equivalence. This is rather a difficult point which will be better appreciated later. The idea is that the scheme of equivalence need not be determinate observationally, and may have permissible transformations; just as the scheme of coordinate-reckoning is not determinate observationally and is subject to transformations. Let $PP_{1}$ represent the displacement $A^{\mu} = \delta x_{\mu}$ which on parallel displacement to~$P'$ becomes~$P'P_{1}'$; then by~\Eq{(91.1)} the difference of coordinates of~$P_{1}'$ and~$P_{1}$ is \[ A^{\mu} + dA^{\mu} = \delta x_{\mu} - \Gamma_{\nu\alpha}^{\mu}\, \delta x_{\alpha}\, dx_{\nu}, \] so that the coordinates of~$P_{1}'$ relative to~$P$ are \[ dx_{\mu} + \delta x_{\mu} - \Gamma_{\nu\alpha}^{\mu}\, \delta x_{\alpha}\, dx_{\nu}. \Tag{(91.2)} \] \PageSep{214} Interchanging the two displacements, i.e.\ displacing $PP'$ along~$PP_{1}$ we shall not arrive at the same point~$P_{1}'$ unless \[ \Gamma_{\nu\alpha}^{\mu} = \Gamma_{\alpha\nu}^{\mu}. \Tag{(91.3)} \] When \Eq{(91.3)}~is satisfied we have the parallelogram law, that if a displacement $AB$ is equivalent to~$CD$, then $AC$~is equivalent to~$BD$. This is the necessary condition for what is called \emph{affine geometry}. It is \index{Affine geometry}% \index{Geometry, Riemannian!affine geometry}% adopted by Weyl and other writers; but J.~A. Schouten in a purely geometrical investigation has dispensed with it. I shall adopt it here. All questions of the fundamental axioms of a science are difficult. In general we have to start somewhat above the fundamental plane and develop the theory backwards towards fundamentals as well as forwards to results. I shall defer until \SecRef{98} the examination of how far the axiom of parallel displacement and the condition of affine geometry are essential in translating the properties of a relation-structure into mathematical expression; and I proceed at once to develop the consequences of the specification here introduced. By the symmetry condition the number of independent~$\Gamma_{\nu\alpha}^{\mu}$ is reduced to~$40$, variable from point to point of space. They are descriptive of the relation-structure of the world, and should contain all that is relevant to physics. Our immediate problem is to show how the more familiar variables of physics can be extracted from this crude material. \Section{92.}{Displacement round an infinitesimal circuit} \index{Parallelogram-law}%