The calculation of the lag in phase caused by Fresnel's etherial motion may proceed thus:—A dense slab of thickness z, which would naturally be traversed with the velocity V/μ, is traversed with the velocity (V/μ) cos ε + (v/μ²) cos θ; where v is the relative velocity of the ether in its neighbourhood; whence the time of journey through it is
μz / V(cos ε + α/μ cos θ), instead of μz / V,
So the equivalent air thickness, instead of being (μ − 1)z, is
μz / cos ε + α/μ cos θ − z = (μ cos ε − α cos θ / (1 − α/μ)² − 1)z,
or, to the first order of minutiæ,
(μ − 1)z − αz cos θ;
θ being the angle between ray and ether drift inside the medium.
So the extra equivalent air layer due to the motion is approximately ±α z cos θ, a quantity independent of μ.
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 quantities, as shown above.
Direct experiments made by Hoek,[11] and by Mascart, on the effect of introducing tubes of water into the path of half-beams of light, are in entire accord with this negative conclusion.