The first restriction is not imperatively demanded, and we shall discard it in Part~II of this chapter. It has the following effect. Equation~\Eq{(84.3)} becomes \begin{align*} \delta(l^{2}) &= F_{\nu\sigma} · g_{\mu\epsilon} A^{\mu} A^{\epsilon} · dS^{\nu\sigma} \\ &= F_{\nu\sigma} l^{2}\, dS^{\nu\sigma}, \intertext{so that} \frac{\delta l}{l} &= \tfrac{1}{2} F_{\nu\sigma}\, dS^{\nu\sigma}. \Tag{(84.4)} \end{align*} The change of length is proportional to the original length and is independent of the direction of the vector; whereas in the more general formula~\Eq{(84.3)} the change of length depends on the direction. One result of the restriction is that zero-length is still zero-length after parallel displacement round a circuit. If we have identified zero-length at one point of the world we can transfer it without ambiguity to every other point and so identify zero-length everywhere. Finite lengths cannot be transferred without ambiguity; a route of parallel displacement must be specified. Zero-length is of great importance in optical phenomena, because in \index{Zero-length of light tracks}% Einstein's geometry any element of the track of a light-pulse is a vector of zero-length; so that if there were no definite zero-length a pulse of light would not know what track it ought to take. It is because Weyl's theory makes no attempt to re-interpret this part of Einstein's theory that an absolute zero-length is required, and the restriction~\Item{(a)} is therefore imposed. Another result of the restriction is that lengths at the same point but in different orientations become comparable without ambiguity. The ambiguity is limited to the comparison of lengths at different places. \PageSep{200} \Section{85.}{Transformation of gauge-systems} According to the foregoing section it is not possible to compare lengths (except zero-length) at different places, because the result of the comparison will depend on the route taken in bringing the two lengths into juxtaposition. In Riemannian geometry we have taken for granted this possibility of comparing lengths. The interval at any point has been assigned a definite value, which implies comparison with a standard; it did not occur to us to question how this comparison at a distance could be made. We have now to define the geometry of the continuum in a way which recognises this difficulty. We suppose that a definite but arbitrary \emph{gauge-system} has been adopted; \index{Gauge-system}% that is to say, at every point of space-time a standard of interval-length has been set up, and every interval is expressed in terms of the standard at the point where it is. This avoids the ambiguity involved in transferring intervals from one point to another to compare with a single standard.