h = 1, 2, 3, 4
Now on the basis of the remarks already made, it is clear that the value of N + δN, when the value of the parameter is λ, will be:—
(10) N + δN = ∫∫∫∫ ((νd(τ + δτ))/dτ)dx dy dz dt,
the integration extending over the whole sichel d(τ + δτ) where d(τ + δτ) denotes the magnitude, which is deduced from
√(-(dx₁ + dδx₁)² - (dx₂ + dδx₂)² - (dx₃ + dδx₃)² - (dx₄ + dδx₄)²)
by means of (9) and
dx₁ = ω₁ dτ, dx₂ = ω₂ dτ,
dx₃ = ω₃ dτ, dx₄ = ω₄ dτ, dλ = 0
therefore:—
(11) (d(τ + δτ))/dτ = √( -∑(ωh + ∑(∂δxh/∂xk)ωk)²)