It must be borne in mind that by this identification the electromagnetic force becomes expressed in some natural unit whose relation to the \CGS\ system is at present unknown. For example the constant of proportionality in~\Eq{(77.7)} may be altered. $F_{\mu\nu}$~is not altered by any change of gauge-system~\Eq{(85.6)} so that its value is a pure number. The question then arises, How many volts per centimetre correspond to $F_{\mu\nu} = 1$ in any given coordinate-system? The problem is a difficult one, but we shall give a rough and rather dubious estimate in \SecRef{102}. I do not think that our subsequent discussion will add anything material to the present argument in favour of the electromagnetic interpretation of~$\kappa_{\mu}$. The case rests entirely on the apparently significant fact, that on removing an artificial restriction in Riemannian geometry, we have just the right number of variables at our disposal which are necessary for a physical description of the world. \Section{86.}{Gauge-invariance} \index{In- (prefix)}% It will be useful to discover tensors and invariants which, besides possessing their characteristic properties with regard to transformations of coordinates, are unaltered by any transformation of gauge-system. These will be called \emph{in-tensors} and \emph{in-invariants}. \index{In-tensors}% There are other tensors or invariants which merely become multiplied by a power of~$\lambda$, when the gauge is altered. These will be called \emph{co-tensors} and \emph{co-invariants}. Change of gauge is a generalisation of change of unit in physical equations, the unit being no longer a constant but an arbitrary function of position. We have only one unit to consider---the unit of interval. Coordinates are merely identification-numbers and have no reference to our unit, so that a displacement~$dx_{\mu}$ is an in-vector. It should be noticed that if we change the unit-mesh of a rectangular coordinate-system from one mile to one kilometre, we make a change of coordinates not a change of gauge. The distinction is more obvious when coordinates other than Cartesian are used. The most confusing case is that of Galilean coordinates, for then the special values of the~$g_{\mu\nu}$ fix the length of side of unit mesh as equal to the unit of interval; and it is not easy to keep in mind that the \emph{displacement} between two corners of the mesh is the number~$1$, whilst the \emph{interval} between them is $1$~kilometre. According to~\Eq{(85.6)} the electromagnetic force~$F_{\mu\nu}$ is an in-tensor. $F_{\mu\nu}$~is only a co-tensor, and $F_{\mu\nu} F^{\mu\nu}$~a co-invariant. \PageSep{203}