ζ = φ (v) z ,
where
and
φ (v) = ac / √ (c2 - v2) = a / β
is a function of v.
If we make no assumption about the initial position of the moving system and about the null-point of t, then an additive constant is to be added to the right hand side.
We have now to show, that every ray of light moves in the moving system with a velocity c (when measured in the moving system), in case, as we have actually assumed, c is also the velocity in the stationary system; for we have not as yet adduced any proof in support of the assumption that the principle of relativity is reconcilable with the principle of constant light-velocity.
At a time τ = t = 0 let a spherical wave be sent out from the common origin of the two systems of co-ordinates, and let it spread with a velocity c in the system K. If (x, y, z), be a point reached by the wave, we have
x2 + y2 + z2 = c2t2