Page:PlummerAberration.djvu/7

258 which express the apparent wave-velocity in a moving medium in terms of the absolute, or vice versa. For a vacuum we put W = W' = U, and we have

$\begin{array}{l} w'=w\ \sec\beta(1-a\ \sin\beta)\\ \\a'=(a-\sin\beta)(1-a\ \sin\beta)\\ \\b'=b\ \cos\beta/(1-a\ \sin\beta)\\ \\c'=c\ \cos\beta/(1-a\ \sin\beta).\end{array}$

8. The laws of stellar aberration and of the Doppler effect are at once deducible, as Einstein has shown. For a star situated in the direction making an angle φ with the direction of motion of the observer we have a = - cos φ and a' = - cos φ', where φ' is the apparent direction. Then we see that

$\begin{array}{rl} w'= & w\ \sec\beta(1+\sin\beta\ \cos\phi)\\ \\\cos\phi'= & (\cos\phi+\sin\beta)/(1+\sin\beta\ \cos\phi)\end{array}$

These expressions can, however, be simplified. For the latter equation gives

$\sin\phi'=\cos\beta\ \sin\phi/(1+\sin\beta\ \cos\phi).$

Hence

$w'\sin\phi'=w\ \sin\phi.$

Again,

$\frac{1-\cos\phi'}{1+\cos\phi'}=\frac{(1-\sin\beta)(1-\cos\phi)}{(1+\sin\beta)(1+\cos\phi)}$

or

$\tan\frac{1}{2}\phi'=\sqrt{\frac{U-v}{U+v}}\tan\frac{1}{2}\phi,$|undefined

which puts the law of aberration in an extremely simple form. If we use wave-lengths instead of wave-frequencies we have

$\lambda'/\sin\phi'=\lambda/\sin\phi.$

This form, which connects directly the apparent wave-length with the apparent aberrational position of the star, becomes useless when φ = φ' = 0, but in this case the product of the last two equations gives in the limit

$\lambda'=\sqrt{\frac{U-v}{U+v}}\lambda,$|undefined

which is the new form of Doppler’s principle. It must he remembered that v is the motion of the observer relative to the medium, and that λ depends on the unknown velocity of the source of light relative to the medium. In some cases we may fairly assume that λ is constant, but λ as well as φ is originally unknown and, if the principle of relativity be accepted in its widest extent, remains unknowable.