Page:The principle of relativity (1920).djvu/215

 where α_{1}, α_{2}, α_{2} are the direction-cosines of the inward-drawn normal to the surface-element dS in the Euclidean Sense. We recognise in this the usual expression for the laws of conservation. We denote the magnitudes t^α_{σ} as the energy-components of the gravitation-field.

I will now put the equation (47) in a third form which will be very serviceable for a quick realisation of our object. By multiplying the field-equations (47) with g^{νσ}, these are obtained in the mixed forms. If we remember that

g^{νσ} [part]Γ^α_{μν}/[part]x_{α} = [part]/[part]x_{α} (g^{νσ} Γ^α_{μν}) - [part]g^{νσ}/[part]x_{α} Γ^α_{μν},

which owing to (34) is equal to

[part]/[part]x_{α} (.g^{νσ} Γ^α_{μν}) - g^{νβ} Γ^σ_{αβ} Γ^α_{μν} - g^{σβ} Γ^ν_{βα} Γ^α_{μν},

or slightly altering the notation equal to

[part]/[part]x_{α} (g^{σβ} Γ^α_{μβ}) - g^{mn} Γ^σ_{mβ} Γ^β_{nμ} and ν for m and n?] - g^{νσ} Γ^α_{μβ} Γ^β_{να}.

The third member of this expression cancel with the second member of the field-equations (47). In place of the second term of this expression, we can, on account of the relations (50), put

κ (t^σ_{μ} - 1/2 δ^σ_{μ} t), where t = t^α_{α}