Page:Philosophical Transactions of the Royal Society A - Volume 184.djvu/839

Rh $ \frac{\mu z}{\mathrm{V} \left ( \cos\epsilon + \frac{\alpha}{\mu}\cos\theta \right )}$, instead of $ \begin{matrix} \mu z \\ \mathrm{V} \end{matrix}$,

or the equivalent air thickness, instead of being $$\left (\mu - 1 \right )z$$, is

$\frac{\mu z}{\cos\epsilon + \frac{\alpha}{\mu}\cos\theta} - z = \left ( \frac {\mu\cos\epsilon - \alpha\cos\theta} {1 - \left ( \frac{\alpha}{\mu} \right )^2} - 1 \right ) z$,

or, to the first order of minutiae, $$ \left (\mu - 1 \right )z - \alpha z\cos\theta$$; $$\theta$$ being the angle between ray and ether drift inside the medium.

So the extra equivalent air layer due to the motion is approximately $$\pm\;\alpha z\cos\theta$$, a quantity independent of $$\mu$$.

Hence, no plan for detecting this first-order effect of motion is in any way assisted by the use of dense stationary substances; their extra ether, being stationary, does not affect the lag caused by motion, except indeed in the second order of small quan tities, as shown above.

Direct experiments made by, and by , on the effect of introducing tubes of water into the path of half beams of light, are in entire accord with this negative conclusion.

Thus, then, we find that no general motion of the entire medium can be detected by changes in direction, or in frequency, or in phase; for on none of them has it any appreciable (i.e., first-order) effect even when assisted by dense matter.

The remaining possible effect that may be looked for is a change of energy.

Effect of Motion on Intensity of Radiation in Different Directions.

18. At first sight it looks as if there ought to be an unequal distribution of energy round a source past which the medium is streaming. For when the waves are drifting along, their energy moves too, and it can thus be distributed unsymmetrically round the source.

The energy emitted per second, or the power of the radiation, is

$\mathrm{P} = 4\pi\rho^2 \mathrm{V}q$,

where $$q$$ is the energy per unit volume at distance $$\rho$$ from the wave centre; supposing that radiating power is unaffected by the motion. So at a place $$r$$, $$\theta$$, reckoning from source as origin, and line of drift as initial line (as in fig. 4), since $$r = \rho \left (\cos\epsilon + \alpha\cos\theta \right )$$,