where
ω´ = ωβ(1 - lv/c), l´ = (l - v/c)/(1 - lv/c), m´ = m/β(1 - lv/c), n´ = n/β(1 - lv/c).
From the equation for ω´ it follows:—If an observer moves with the velocity v relative to an infinitely distant source of light emitting waves of frequency ν, in such a manner that the line joining the source of light and the observer makes an angle of Φ with the velocity of the observer referred to a system of co-ordinates which is stationary with regard to the source, then the frequency ν´ which is perceived by the observer is represented by the formula
ν´ = ν (1 - cosΦ(v/c))/[sqrt](1 - v^2/c^2).
This is Döppler's principle for any velocity. If Φ = 0, then the equation takes the simple form
ν´ = ν((1 - v/c)/(1 + v/c))^{1/2}.
We see that—contrary to the usual conception—ν´?] = [infinity], for v = -c.
If Φ´ = angle between the wave-normal (direction of the ray) in the moving system, and the line of motion of the observer, the equation for l´ takes the form
cosΦ´ = (cosΦ - v/c) / (1 - v/c cosΦ).
