v
M′ = β(M₀ - --- Z₀) sin Φ′
c
v
N′ = β(N₀ - --- Y₀) sin Φ′
c
l′ξ + m′η + n′ζ
Φ′ = ω′(t - --------------- )
c
where
$$ \omega' = \omega \beta (1 - \frac {lv}{c}) $$ ,
$$ l' = \frac {l - \frac {v}{c}}{1 - \frac {lv}{c}} $$ ,
$$ m' = \frac {m}{\beta (1 - \frac {lv}{c})} $$ ,
$$ n' = \frac {n}{\beta (1 - \frac {lv}{c})} $$
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
$$ \nu' = \nu \frac {1 - cos \Phi \frac {v}{c}} {\sqrt {1 - \frac
{v^2}{c^2}}} $$
This is Döppler’s principle for any velocity. If Φ = 0, then the
equation takes the simple form
$$ \nu' = \nu (\frac {1 - \frac {v}{c}}{1 + \frac {v}{c}})^{\frac
{1}{2}} $$
We see that—contrary to the usual conception—ν = ∞, 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
$$ \cos \Phi' = \frac {\cos \Phi - \frac {v}{c}} {1 - \frac {v}{c} \cos
\Phi} $$
This equation expresses the law of observation in its most general form.
If Φ = π/2, the equation takes the simple form
v
cos Φ′ = ---
c
We have still to investigate the amplitude of the waves, which occur in
these equations. If A and A′ be the amplitudes in the stationary and the
moving systems (either electrical or magnetic), we have
$$ A'^2 = A^2 \frac {(1 - \frac {v}{c} \cos \Phi)^2} {1 - \frac
{v^2}{c^2}} $$