Page:PlummerAberration.djvu/6

for the relative velocities. Let W be the absolute velocity of a ray which is parallel to the axis of x or ξ in a moving medium which has the refractive index μ when at rest. Then

${\frac {d\xi }{d\tau }}=(W-v)/\left(1-Wv/U^{2}\right).$

But this is μ^-1U if the apparent ray-velocity is unaltered by the motion. Hence

$W-v=\mu ^{-1}U\left(1-Wv/U^{2}\right)$

or

$W=\mu ^{-1}U+\left(1-\mu ^{-2}\right)v/\left(1+\mu ^{-1}v/U\right).$

The second member on the right is the apparent drift of plane light-waves which are normal to the direction of motion. To the first order it agrees with Fresnel’s expression (1 - μ^-2)v and is consistent with Fizeau’s experiment, which cannot be performed accurately enough to verify it to a higher order. For the rest we have a general explanation of the null effect of optical experiments without supposing that the ether is carried along in the neighbourhood of the Earth without relative motion.

7. According to the electromagnetic theory a plane wave of light depends upon two periodic vectors, the components of which contain the factor

$\sin \ 2\pi w\{t-(ax+by+cz)/W\}$,

where a, b, c are the direction cosines of the wave-normal and W is the wave-velocity. To the observer moving with velocity v, the corresponding factor is

$\sin \ 2\pi w'\{\tau -(a'\xi -b'\eta +c'\zeta )/W'\}$.

The transformation of § 4 gives the relations

${\begin{array}{rl}w&=w'(\sec \beta +a'U\ \tan \beta /W')\\\\wa/W&=w'(a'\ \sec \beta /W'+\tan \ \beta /U)\\\\wb/W&=w'b'/W'\\\\wc/W&=w'c'/W'\end{array}}$

or, if the transformation be reversed (which only requires the sign of β to be changed),

${\begin{array}{rl}w'&=w\ \sec \beta (1-aU\sin \beta /W)\\\\a'/W'&=(a/W-\sin \beta /U)/(1-aU\ \sin \beta /W)\\\\b'/W'&=b\ \cos \beta /W(1-aU\ \sin \beta /W)\\\\c'/W'&=c\ \cos \beta /W(1-aU\ \sin \beta /W)\end{array}}$

The latter equations give

$1/W'^{2}=\left\{1-2aW\ \sin \beta /U-\left(1-a^{2}-W^{2}/U^{2}\right)\sin ^{2}\beta \right\}\div W^{2}(1-aU\ \sin \beta /W)^{2},$

or

$1/W{}^{2}=\left\{1+2a'W'\ \sin \beta /U-\left(1-a'^{2}-W'^{2}/U^{2}\right)\sin ^{2}\beta \right\}\div W'^{2}(1+a'U\ \sin \beta /W')^{2},$