= - (ex + imx) sin (φ + iψ) + (ey + imy) cos. (φ + iψ).
§ 4. 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 about this last axis. So we came to a more general law:—
Let v be a vector with the components vx, vy, vz, and let | v | = q < 1. By ṽ we shall denote any vector which is perpendicular to v, and by rv, rṽ we shall denote components of r in direction of ṽ and v.
Instead of (x, y, z, t), new magnetudes (x′ y′ z′ t′) will be introduced in the following way. If for the sake of shortness, r is written for the vector with the components (x, y, z) in the first system of reference, r′ for the same vector with the components (x′ y′ z′) in the second system of reference, then for the direction of v, we have
(10) r′v = (rv - qt)/√(1 - q²)
and for the perpendicular direction ṽ,
(11) r′ṽ = rṽ
and further (12) t′ = (-qrv + t)/√(1 - q²).
The notations (r′ṽ, r′v) are to be understood in the sense that with the directions v, and every direction ṽ perpendicular to v in the system (x, y, z) are always associated the directions with the same direction cosines in the system (x′ y′ z′).