time , if the ray is sent in the direction of increasing ξ, we have
, i.e. .
Now the ray of light moves relative to the origin of k with a velocity c-v, measured in the stationary system ; therefore we have
.
Substituting these values of t in the equation for ξ, we obtain
.
In an analogous manner, we obtain by considering the ray of light which moves along the y-axis,
,
where .
Therefore .
If for x', we substitute its value x—tv, we obtain
- ,
- ,
- ,
- ,
where , and is a function of v.