Page:The principle of relativity (1920).djvu/45

 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.