with the aid of our transformation-equations, let us transform this equation, and we obtain by a simple calculation,
ξ2 + η2 + ζ2 = c2τ2.
Therefore the wave is propagated in the moving system with the same velocity c, and as a spherical wave.[[7]] 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 two-fold application of the transformation-equations, we obtain
v
τ′ = φ(-v)β(-v){τ + --- ξ}
c2
= φ(v)φ(-v)t,
x′ = φ](v)β(v)(ξ + vτ)