Page:The principle of relativity (1920).djvu/192

 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).