Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/138

 The same result may easily be deduced from an experiment performed by Hoek. In this a beam of light was divided into two portions, one of which was made to pass through a tube of water AB and was then reflected at a mirror C, the light being afterwards allowed to return to A without passing through the water: while the other portion of the bifurcated beam was made to describe the same path in the reverse order, i.e. passing through the water on its return journey from C instead of on the outward journey. On causing the two portions of the beam to interfere, Hoek found that no difference of phase was produced between them when the apparatus was oriented in the direction of the terrestrial motion.

Let w denote the velocity of the earth, supposed to be directed from the tube towards the mirror. Let c/μ denote the velocity of light in the water at rest, and c/μ + φ the velocity of light in the water when moving. Let l denote the length of the tube. The magnitude of the distance BC does not affect the experiment, so we may suppose it zero. The time taken by the first portion of the beam to perform its journey is evidently

while the time for the second portion of the beam is

The equality of these expressions gives at once, when terms of higher orders than the first in w/c are neglected,

which is Fresnel's expression.