"(A)."
On the other hand the law of transformation of the volume element
dτ′ = ∫ dx₁ dx₂ dx₃ dx₄
is according to the wellknown law of Jacobi.
"(B)."
by multiplication of the two last equations (A) and (B) we get
(18) = √g dτ′ = √g dτ.
Instead of √g, we shall afterwards introduce √(-g) which has a real value on account of the hyperbolic character of the time-space continuum. The invariant √(-g)dτ, is equal in magnitude to the four-dimensional volume-element measured with solid rods and clocks, in accordance with the special relativity theory.
Remarks on the character of the space-time continuum—Our assumption that in an infinitely small region the special relativity theory holds, leads us to conclude that ds² can always, according to (1) be expressed in real magnitudes dX₁ ... dXh. If we call dτ₀ the “natural” volume element dX₁ dX₂ dX₃ dX₄ we have thus (18a) dτ₀ = √(g)iτ.