δ(_dx__{ν}/_d_λ) = (_d_/_d_λ)(δ_x__{ν})
after partial integration,
{ λ₃
{ ∫ _d_λ _k__{σ} δ_x__{σ} = 0
(20b) { λ₁
{
{ where _k__{σ} = (_d_/_d_λ){(_g__{μν}/ω) · (_dx__{μ}/_d_λ)}
- (1/(2ω))(∂_g__{μν}/∂_x__{σ}
× (_dx__{μ}/_d_λ) · (_dx__{ν}/_d_λ).
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)
$$ g_{\mu\nu} \frac{\partial^2 x_{\mu}}{\partial s^2} + \frac{\partial
g_{\mu\nu}}{\partial x_{\sigma}} \frac{\partial x_{\sigma}}{\partial s}
\frac{\partial x_{\mu}}{\partial s} - \frac{1}{2} \frac{\partial
g_{\mu\sigma}}{\partial x_{\nu}} \frac{\partial x_{\mu}}{\partial s}
\frac{\partial x_{\sigma}}{\partial s} = 0 $$
Or by merely changing the notation suitably,
(20d) $$ g_{\alpha\sigma} \frac{d^2 x_{\alpha}}{ds^2} +
\begin{bmatrix}\mu\nu\\\sigma\end{bmatrix} \frac{dx_{\mu}}{ds}
\frac{dx_{\nu}}{ds} = 0 $$
where we have put, following Christoffel,
(21)
$$ \begin{bmatrix}\mu\nu\\\sigma\end{bmatrix} = \frac{1}{2}
\begin{bmatrix}\frac{\partial g_{\mu\sigma}}{\partial x_{\nu}} +
\frac{\partial g_{\nu\sigma}}{\partial x_{\mu}} - \frac{\partial
g_{\mu\nu}}{\partial \sigma}\end{bmatrix} $$
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—
$$ \frac{d^2 x_{\tau}}{ds^2} + \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix}
\frac{dx_{\mu}}{ds} \frac{dx_{\nu}}{ds} = 0 $$
Here we have put, following Christoffel,
$$ \begin{Bmatrix}\mu\nu\\\tau\end{Bmatrix} = g^{\tau\alpha}
\begin{bmatrix}\mu\nu\\\alpha\end{bmatrix} $$
§ 10. Formation of Tensors through Differentiation.
Relying on the equation of the geodetic line, we can now easily deduce
laws according to which new tensors can be formed from given tensors by
differentiation. For this purpose, we would first establish the general
co-variant differential equations. We achieve this through a repeated
application of the following simple law. If a certain curve be given in
our continuum whose points are characterised by the arc-distances _s_,
measured from a fixed point on the curve, and if further φ, be an
invariant space function, then _d_φ/_ds_ is also an invariant. The proof
follows from the fact that _d_φ as well as _ds_, are both invariants