Page:The principle of relativity (1920).djvu/227

 On account of (60) the second member on the right-hand side admits of the transformation—

F^{μν} [part]F_{σμ}/[part]x_{ν} = -1/2 F^{μν} [part]F_{μν}/[part]x_{σ}

= -1/2 g^{μα} g^{νβ} F_{αβ} [part]F_{μν}/[part]x_{σ}.

Owing to symmetry, this expression can also be written in the form

= -1/4 [g^{μα} g^{νβ} F_{αβ} [part]F_{μν}/[part]x_{σ}

+ g^{μα} g^{νβ} [part]F_{αβ}/[part]x_{σ} F_{μν}],

which can also be put in the form

- 1/4 [part]/[part]x_{σ} (g^{μα} g^{νβ} F_{αβ} F_{μν})

+ 1/4 F_{αβ} F_{μν} [part]/[part]x_{σ} (g^{μα} g^{νβ}).

The first of these terms can be written shortly as

- 1/4 [part]/[part]x_{σ} (F^{μν} F_{μν}),

and the second after differentiation can be transformed in the form

- 1/2 F^{μτ} F_{μν} g^{νρ} [part]g_{στ}/[part]x_{σ}.