Page:PlummerAberration.djvu/7

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^[1] 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

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

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

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

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.

1. Ann. der Physik, 1905, No. 10, p. 891.