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
x^2 + y^2 + z^2 = c^{2}t^{2}.
with the aid of our transformation-equations, let us transform this equation, and we obtain by a simple calculation,
ξ^2 + η^2 + ζ^2 = c^{2}τ^{2}.
Therefore the wave is propagated in the moving system with the same velocity c, and as a spherical wave.[1] Therefore we show that the two principles are mutually reconcilable.
In the transformations we have got an undetermined function φ(v), and we now proceed to find it out.
Let us introduce for this purpose a third co-ordinate system k´, which is set in motion relative to the system k, the motion being parallel to the ξ-axis. Let the velocity of the origin be (-v). At the time t = 0, all the initial co-ordinate points coincide, and for t = x = y = z = 0, the time t´ of the system k´ = 0. We shall say that (x´, y´, z´, t´) are the co-ordinates measured in the system k´, then by a
- ↑ Vide Note 9.
