Page:The principle of relativity (1920).djvu/203

 If we replace g^{μα} g^{νβ} A_{μνσ} by A_σ^{αβ}, g^{μα} g^{νβ} A_{μν} by A^{αβ} and replace in the transformed first member

[part]g^{νβ}/[part]x_{σ} and [part]g^{μα}/[part]x_{σ}

with the help of (34), then from the right-hand side of (27) there arises an expression with seven terms, of which four cancel. There remains

(38) A_{σ}^{αβ} = [part]A^{αβ}/[part]x_{σ} + {σ κ / α} A^{κβ} + {σ κ / β} A^{ακ}.

This is the expression for the extension of a contravariant tensor of the second rank; extensions can also be formed for corresponding contravariant tensors of higher and lower ranks.

We remark that in the same way, we can also form the extension of a mixed tensor A_{μ}^{α}

(39) A_{μσ}^{α} = [part]A_{μ}^{α}/[part]x_{σ} - {σ μ / τ} A_{τ}^{α} + {σ τ / α} A_{μ}^{τ}.

By the reduction of (38) with reference to the indices β and σ(inner multiplication with δ_{β}^{σ}), we get a contravariant four-vector

A^{α} = [part]A^{αβ}/[part]x_{β} + {β κ / β} A^{ακ} + {β κ / α} A^{κβ}.