Page:The principle of relativity (1920).djvu/85

 about this last axis. So we came to a more general law:—

Let v be a vector with the components v_{x}, v_{y}, v_{z}, and let | v | = q < 1. By [=v] we shall denote any vector which is perpendicular to v, and by r_{v}, r_{[=v]} we shall denote components of r in direction of [=v] 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} = (r_{v} - qt)/[sqrt](1 - q^2)

and for the perpendicular direction [=v],

(11) r´_{[=v]} = r_{[=v]}

and further (12) t´ = (-qr_{v} + t)/[sqrt](1 - q^2).

The notations (r´_{[=v]}, r´_{v}) are to be understood in the sense that with the directions v, and every direction [=v] 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´).

A transformation which is accomplished by means of (10), (11), (12) with the condition 0 < q < 1 will be called a special Lorentz-transformation. We shall call v the vector, the direction of v the axis, and the magnitude of v the moment of this transformation.

If further ρ´ and the vectors u´, e´, m´, in the system (x´ y´ z´) are so defined that,

(13) ρ´ = ρ[(-qu_{v} + 1)/[sqrt](1 - q^2)], ρ´u´_{v} = ρ(u_{v} - q)/[sqrt](1 - q^2), ρ´u_{[=v]} =, ρ´u_{v},