Page:The principle of relativity (1920).djvu/206

 it a new tensor, namely the extension of the fundamental tensor. We ean easily convince ourselves that this vanishes identically. We prove it in the following way; we substitute in (27)

A_{μυ} = [part]A_{μ}/[part]x_{υ} - {μυ ρ}A_{ρ} should be ν?]

i.e., the extension of a four-vector.

Thus we get (by slightly changing the indices) the tensor of the third rank

A_{μστ} = [part]^2 A_{μ}/([part]x_{σ}[part]x_{τ}) - {μσ ρ} [part]A_{ρ}/[part]x_{τ} - {μτ ρ} [part]A_{ρ}/[part]x_{σ} - {στ ρ} [part]A_{μ}/[part]x_{ρ}

+ [-[part]/[part]x_{τ} {μσ ρ} + {μτ α}{ασ ρ} + {στ α}{αμ ρ}]A_{ρ}.

We use these expressions for the formation of the tensor A_{μστ} - A_{μτσ}. Thereby the following terms in A_{μστ} cancel the corresponding terms in A_{μτσ}; the first member, the fourth member, as well as the member corresponding to the last term within the square bracket. These are all symmetrical in σ, and τ. The same is true for the sum of the second and third members. We thus get

A_{μστ} - A_{μτσ} = B^ρ_{μστ} A_{ρ}.

{B^ρ_{μστ} = -[part]/[part]x_{τ} {μσ ρ} + [part]/[part]x_{σ} {μτ ρ} (43){   {- {μσ α}{ατ ρ} + {μτ α}{ασ ρ}