From which it follows, since the choice of δν_{σ} is perfectly
arbitrary that k_{σ}'s should vanish; Then
(20c) k_σ = 0 (σ = 1, 2, 3, 4)
are the equations of geodetic line; since along the geodetic line considered we have ds [/=] 0, we can choose the parameter λ, as the length of the arc measured along the geodetic line. Then w] = 1, and we would get in place of (20c)
g_{μν} [part]^2x_μ/[part]s^2 + [part]g_{μν}/[part]x_σ [part]x_σ/[part]s [part]x_μ/[part]s
- 1/2 [part]g_{μσ}/[part]x_ν [part]x_μ/[part]s [part]x_σ/[part]s = 0.
Or by merely changing the notation suitably,
(20d) g_{ασ} d^2x_α/ds^2 + [^{μν} _{σ}] dx_μ/ds · dx_ν/ds = 0
where we have put, following Christoffel,
(21) [^{μν} _{σ}] = 1/2 [[part]g_{μσ}/[part]x_ν + [part]g_{νσ}/[part]x_μ - [part]g_{μν}/[part]σ].
Multiply finally (20d) with g^{στ} (outer multiplication with reference to τ, and inner with respect to σ) we get at last the final form of the equation of the geodetic line—
d^2x_τ/ds^2 + {^{μν} _{τ}} dx_μ/ds · dx_ν/ds = 0.
Here we have put, following Christoffel,
{^{μν} _{τ}} = g^{τα} [^{μν} _{α}].
