Page:The principle of relativity (1920).djvu/97

 (x, y, z, t)—the neighbourhood may be supposed to be in motion in any possible manner, then for the space-time point x, y, z, t, the same relations (A) (B) (V) which hold in the case when all matter is at rest, shall also hold between ρ, the vectors C, e, m, M, E and their differentials with respect to x, y, z, t. The second axiom shall be:—

Every velocity of matter is < 1, smaller than the velocity of propogation of light.[A]

The fundamental equations are of such a kind that when (x, y, z, it) are subjected to a Lorentz transformation and thereby (m - ie) and (M - iE) are transformed into space-time vectors of the second kind, (C, iρ) as a space-time vector of the 1st kind, the equations are transformed into essentially identical forms involving the transformed magnitudes.

Shortly I can signify the third axiom as:—

(m, -ie), and (M, -iE) are space-time vectors of the second kind, (C, ip) is a space-time vector of the first kind.

This axiom I call the Principle of Relativity.

In fact these three axioms lead us from the previously mentioned fundamental equations for bodies at rest to the equations for moving bodies in an unambiguous way.

According to the second axiom, the magnitude of the velocity vector | u | is < 1 at any space-time point. In consequence, we can always write, instead of the vector u, the following set of four allied quantities

ω_{1} = u_{x}/[sqrt](1 - u[ ** F1: not Italic?]^2), ω_{2} = u_{y}/[sqrt](1 - u^2), ω_{3} = u_{z}/[sqrt](1 - u^2), ω_{4} = i/[sqrt](1 - u^2)

[Footnote: A Vide Note.]