,
,
, that is, concerning particular metrical properties of the manifold that go beyond the postulate of the freedom of movement of the measuring rods. But, if we demand, in particular, that each point of the manifold may be fixed by means of rectangular Cartesian co-ordinates, whereby particular assumptions are made concerning the possible ways of placing the measuring rods, then the line-element, expressed in these special variables, assumes the form
Hitherto this expression has always been introduced for the length of the line-element in all physical laws. It is contained in the more general expression of Riemann's line-element
as the special case
