on the other hand, preserves its form only for transformations of the special theory of relativity, which confine themselves to systems moving uniformly and rectilinearly. Consequently, the element of arc must be adapted to the further requirements of a general theory of relativity so that it preserves its form after any substitutions whatsoever.
The choice of the expression
to represent the line-element in physical laws is, in spite of its very general character, still to be regarded as a hypothesis, as Riemann has already pointed out. For there are other functions of the differentials
,
,
