(6) e´_{x´} = (e_{x} - qm_{y})/([sqrt](1 - q^2)), m´_{r´} = (qe_{x} + m_{y})/([sqrt](1 - q^2)), e´_{z´} = e_{z}.
(7) m´_{x´} = (m_{x} - qe_{y})/([sqrt](1 - q^2)), e´_{y´} = (qm_{x} + e_{y})/([sqrt](1 - q^2)), m´_{z´} = m_{z}.
Then we have for these newly introduced vectors u´, e´, m´ (with components u_{x}´, u_{y}´, u_{z}´; e_{x}´, e_{y}´, e_{z}´; m_{x}´, m_{y}´, m_{z}´), and the quantity ρ´ a series of equations I´), II´), III´), IV´) which are obtained from I), II), III), IV) by simply dashing the symbols.
We remark here that e_{x} - qm_{y}, e_{y} + qm_{x} are components of the vector e + [vm], where v is a vector in the direction of the positive Z-axis, and | v | = q, and [vm] is the vector product of v and m; similarly -qe_{x} + m_{y}, m_{x} + qe_{y} are the components of the vector m - [ve].
The equations 6) and 7), as they stand in pairs, can be expressed as.
e´_{x´} + im´_{x´} = (e_{x} + im_{x}) cos iψ + (e_{y} + im_{y}) sin iψ,
e´_{y´} + im´_{y´} = - (e_{x} + im_{x}) sin iψ + (e_{y} + im_{y}) cos iψ,
e´_{z´} + im´_{z´} = e´_{z} + im_{z}.
If φ denotes any other real angle, we can form the following combinations:—
(e´_{x´} + im´_{x´}) cos. φ + (e´_{y´´} + im´_{y´}) sin φ
= (e_{x} + im_{x}) cos. (φ + iψ) + (e_{y} + im_{y}) sin (φ + iψ),
= (e´_{x´} + im´_{x´}) sin φ + (e´_{y´} + im´_{y´}) cos. φ
= - (e_{x} + im_{x}) sin (φ + iψ) + (e_{y} + im_{y}) cos. (φ + iψ).
Special Lorentz Transformation.
The rôle which is played by the Z-axis in the transformation (4) can easily be transferred to any other axis when the system of axes are subjected to a transformation
