time τ = 0, if the ray is sent in the direction of increasing ξ, we have
ξ = cτ, i.e. ξ = ac(t - vx´/(c^2 - v^2)).
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
x´/(c - v) = t.
Substituting these values of t in the equation for ξ, we obtain
ξ = a(c^2/(c^2 - v^2))x´.
In an analogous manner, we obtain by considering the ray of light which moves along the y-axis,
η = cτ = ac(t - vx´/(c^2 - v^2)),
where y/[sqrt](c^2 - v^2) = t, x´ = 0,
Therefore η = a(c/[sqrt](c^2 - v^2))y, ζ = a(c/[sqrt](c^2 - v^2))z.
If for x´, we substitute its value x - tv, we obtain
τ = φ (v). β(t - vx/c^2),
ξ = φ (v). β(x - vt),
η = φ (v) y
ζ = φ (v) z,
where β = 1/[sqrt](1 - v^2/c^2), and φ (v) = ac/[sqrt](c^2 - v^2) = a/βa P2: I agree with the correction] is a function of v.
