Page:The principle of relativity (1920).djvu/98

 with the relation

(27) ω_{1}^2 + ω_{2}^2 + ω_{3}^2 + ω_{4}^2 = - |

From what has been said at the end of § 4, it is clear that in the case of a Lorentz-transformation, this set behaves like a space-time vector of the 1st kind.

Let us now fix our attention on a certain point (x, y, z) of matter at a certain time (t). If at this space-time point u = 0, then we have at once for this point the equations (A), (B) (V) of § 7. It u [/=] 0, then there exists according to 16), in case | u | < 1, a special Lorentz-transformation, whose vector v is equal to this vector u (x, y, z, t), and we pass on to a new system of reference (x´ y´ z´ t´) in accordance with this transformation. Therefore for the space-time point considered, there arises as in § 4, the new values 28) ω´_{1} = 0, ω´_{2} = 0, ω´_{3} = 0, ω´_{4} = i, therefore the new velocity vector ω´ = 0, the space-time point is as if transformed to rest. Now according to the third axiom the system of equations for the transformed point (x´ y´ z´ t´) involves the newly introduced magnitude (u´ ρ´, C´, e´, m´, E´, M´) and their differential quotients with respect to (x´, y´, z´, t´) in the same manner as the original equations for the point (x, y, z, t). But according to the first axiom, when u´ = 0, these equations must be exactly equivalent to

(1) the differential equations (A´), (B´), which are obtained from the equations (A), (B) by simply dashing the symbols in (A) and (B).

(2) and the equations

(V´) e´ = E´, M' = μm´, C´ = σE´

where, μ, σ are the dielectric constant, magnetic permeability, and conductivity for the system (x´ y´ z´ t´) i.e. in the space-time point (x y, z t) of matter.