Page:The principle of relativity (1920).djvu/253

 Notes 13 and 14.

We have denoted the four-vector ω by the matrix the reciprocal matrix
 * ω_{1} ω_{2} ω_{3} ω_{4} |. It is then at once seen that ω̄ denotes


 * ω_{1} |
 * ω_{2} |
 * ω_{3} |
 * ω_{4} |

It is now evident that while ω^1 = ωA, ω̄^1 = A^{-1}ω̄

[ω, s] The vector-product of the four-vector ω and s may be represented by the combination

[ωs] = ω̄s - [=s]ω

It is now easy to verify the formula [function]^1 = A^{-1}[function]A. Supposing for the sake of simplicity that [function] represents the vector-product of two four-vectors ω, s, we have

[function]^1 = [ω^1s^1] = [ω̄^1s^1 - [=s]^1ω^1]

= [A^{-1} ω̄sA - A^{-1}sω̄A]

= A^{-1}[ω̄s - sω̄]A = A^{-1}[function]A.

Now remembering that generally

[function] = ρφ + ρ*φ*.

Where ρ, ρ* are scalar quantities, φ, φ* are two mutually perpendicular unit planes, there is no difficulty in seeming that

[function]^1 = A^{-1}[function]A.

Note 15. The vector product (w[function]). (P. 36).

This represents the vector product of a four-vector and a six-vector. Now as combinations of this type are of