Page:The principle of relativity (1920).djvu/184

 We now prove a law, which will be often applicable for proving the tensor-character of certain quantities. According to the above representation, A_{μν} B^{μν} is a scalar, when A_{μν} and B^{στ}ν}?] are tensors. We also remark that when A_{μν} B^{μν} is an invariant for every choice of the tensor B^{μν}, then A_{μν} has a tensorial character.

Proof:—According to the above assumption, for any substitution we have

A_{στ´} B^{στ´} = A_{μν} B^{μν}.

From the inversion of (9) we have however

B^{μν} = ([part]x_{μ}/[part]x_{σ´}) ([part]x_{ν}/[part]x_{τ´}) B^{στ´}.

Substitution of this for B^{μν} in the above equation gives

(A_{στ´} - ([part]x_{μ}/[part]x_{σ´}) ([part]x_{ν}/[part]x_{τ´}) A_{μν}) B^{στ´} = 0.

This can be true, for any choice of B^{στ´} only when the term within the bracket vanishes. From which by referring to (11), the theorem at once follows. This law correspondingly holds for tensors of any rank and character. The proof is quite similar, The law can also be put in the following from. If B^{μ} and C^{ν} are any two vectors, and