Page:SahaElectrodynamics.djvu/23

 where

$$\omega'=\omega\beta\left(1-\frac{lv}{c}\right),\ l'=\frac{l-\frac{v}{c}}{1-\frac{lv}{c}},\ m'=\frac{m}{\beta\left(1-\frac{lv}{c}\right)},\ n'=\frac{n}{\beta\left(1-\frac{lv}{c}\right)}$$.

From the equation for $$\omega'$$ it follows :— If an observer moves with the velocity v relative to an infinitely distant source of light emitting waves of frequency $$\nu$$, in such a manner that the line joining the source of light and the observer makes an angle of $$\Phi$$ with the velocity of the observer referred to a system of co-ordinates which is stationary with regard to the source, then the frequency $$\nu'$$ 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 Doppler's principle for any velocity. If $$\Phi=0$$ then the equation takes the simple form

$$\nu'=\nu\left(\frac{1-\frac{v}{c}}{1+\frac{v}{c}}\right)^{\frac{1}{2}}$$.

We see that —contrary to the usual conception— $$\nu=\infty$$, for $$v = -c$$.

If $$\Phi'$$=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

$$\cos\Phi'=\frac{\cos\Phi-\frac{v}{c}}{1-\frac{v}{c}\cos\Phi}$$.