_c²_ ξ = _a_ ------------- _x′_ _c²_ - _v²_ In an analogous manner, we obtain by considering the ray of light which moves along the _y_-axis, _vx′_ η = _c_τ = _a c_(_t_ - ------------- ) _c²_ - _v²_ where _y_ ------------------ = _t_, _x′_ = 0, √ (_c²_ - _v²_) Therefore _c_ η = _a_ ------------------ _y_, √ (_c²_ - _v²_) _c_ ζ = _a_ ----------------- _z_ . √ (_c²_ - _v²_) If for _x′_, we substitute its value _x_ - _tv_, we obtain _v_._c_ τ = φ (_v_). β (_t_ - ----------- , c² ξ = φ (_v_). β (_x_ - _vt_) , η = φ (_v_) _y_ ζ = φ (_v_) _z_ , where $$ \beta = \frac {1}{\sqrt {1 - \frac {v^2}{c^2}}} $$ 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_ = _c²__t²_ with the aid of our transformation-equations, let us transform this equation, and we obtain by a simple calculation, ξ² + η² + ζ² = _c²_τ². 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.