Page:The principle of relativity (1920).djvu/198

 It would therefore be sufficient to deduce the expression of extension, for one such special tensor. According to (26) we have the expressions

[part]A_{μ}/[part]x_{σ} - {^{σμ} _{τ}} A_{τ}

[part]B_{ν}/[part]x_{σ} - {^{σν} _{τ}} B_{τ}

are tensors. Through outer multiplication of the first with B_{ν} and the 2nd with A_{μ} we get tensors of the third rank. Their addition gives the tensor of the third rank

A_{μνσ} = [part]A_{μν}/[part]x_{σ} - {^{σμ} _{τ}} A_{τν} - {^{σν} _{τ}} A_{μτ} (27)

where A_{μν} is put = A_{μ} B_{ν}. The right hand side of (27) is linear and homogeneous with reference to A_{μν}, and its first differential co-efficient so that this law of formation leads to a tensor not only in the case of a tensor of the type A_{μ} B_{ν} but also in the case of a summation for all such tensors, i.e., in the case of any co-variant tensor of the second rank. We call A_{μνσ} the extension of the tensor A_{μν}. It is clear that (26) and (24) are only special cases of (27) (extension of the tensors of the first and zero rank). In general we can get all special laws of formation of tensors from (27) combined with tensor multiplication.