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sign δ corresponds to a passage from a point of the geodetic curve sought-for to a point of the contiguous curve, both lying on the same surface λ.
Then (20) can be replaced by
{ λ_3
{ [integral]δω dλ = 0
(20a) { λ_1
{
{ ω^2 = g_{μν}(dx_μ/dλ)(dx_ν/dλ)
But
δω = (1/ω){1/2([part]g_{μν}/[part]x_σ) · (dx_μ/dλ) · (dx_ν/dλ) · δx_σ
+ g_{μν}(dx_μ/dλ)δ(dx_ν/dλ)}
So we get by the substitution of δω in (20a), remembering that
δ(dx_ν/dλ) = (d/dλ)(δx_ν)
after partial integration,
{ λ_3
{ [integral] dλ k_σ δx_σ = 0
(20b) { λ_1
{
{ where k_σ = (d/dλ){(g_{μν}/ω) · (dx_μ/dλ)} - (1/(2ω))([part]g_{μν}/[part]x_σ
×(dx_μ/dλ) · (dx_ν/dλ).
