is a co-variant four-vector (gradient of φ).
According to our law, the differential-quotient χ = ∂ψ/∂s taken along any curve is likewise an invariant.
Substituting the value of ψ, we get
Here however we can not at once deduce the existence of any tensor. If we however take that the curves along which we are differentiating are geodesics, we get from it by replacing d²xν/ds² according to (22)
From the interchangeability of the differentiation with regard to μ and ν, and also according to (23) and (21) we see that the bracket
is symmetrical with respect to μ and ν.
As we can draw a geodetic line in any direction from any point in the continuum, ∂xμ/ds is thus a four-vector, with an arbitrary ratio of components, so that it follows from the results of §7 that