The Principle of Relativity

by Albert Einstein; H. Minkowski

multiplying these equations by v_x^* . u_x^*, or by v_y^* . u_y^*, we obtain

φ_{x z}^* φ_{x l} + φ_{y z}^* φ_{y l} = 0 and φ_{x y}^* φ_{x l} + φ_{z x}^* φ_{z l} = 0

from which we have

φ_{y z}^* : φ_{x y}^* : φ_{z x}^* = φ_{x l} : φ_{z l} : φ_{y l}

In a corresponding way we have

φ_{y z} : φ_{x y} : φ_{z x} = φ_{x l}^* : φ_{z l}^* : φ_{y l}^*.

i.e. φ_{i k}^* = λφ(_{i k})

when the subscript (ik) denotes the component of φ in the plane contained by the lines other than (ik). Therefore the theorem is proved.

We have (φ φ*) = φ_{y z} φ_{y z}^* + ...

= 2 (φ_{y z} φ_{z l} + ...)