Searching for the Ether
In this specific case, the contraction of the Earth could produce no effect, because both measurements were made relative to the same reference (the local vertical) and the surface of the mercury mirror is, of course, perpendicular to the local vertical, whatever the changes that the gravitational field could undergo due to Lorentz contraction. The predicted effect was a small systematic difference between the direct and the reflected angles, which should depend on the direction of the observatory relative to the motion of the Earth through the ether.
Fig. 5. Elamack’s diagram for analyzing the reflection of light in a moving mirror. The initial position of the mirror is S, and after a time 81 its position is S'. AA' is a wave front of the incident light beam, and BE is a wave front of the reflected beam.
Let 9 be the angle of incidence and 9' the angle of reflection of a light ray in a moving mirror, measured relative to the ether (Fig. 5).10 According to Eiamack's analysis, instead of 9=9' the following equations would hold:
sin 9' = (1 - P2) sin 9 / (1 + 2P cos 9 + P2) (4)
cos 9' = [(1 + P2) cos 9 + 2P] / (1 + 2P cos 9 + P2)(5)
16 In his equations Courvoisier used 9 as a symbol of sidereal time, but in this particular derivation we are following Hamack's notation in his paper “Zur Theorie des bewegten Spiegels” (ref. 15).
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Roberto Martins Searching for the Ether DIO 17
In those equations, the speed of the mirror is P=v/c, in the direction perpendicular to the mirror. Any motion of the mirror parallel to its surface would have no influence upon the direction of light. In the case of the mercury mirror, the relevant direction if the local vertical, and therefore p, here, has the same general meaning ascribed by Courvoisier to this symbol. Relative to the proper reference system of the mirror there is an aberration effect, and the angles of incidence (z) and reflection (z1) are:
z = 9 + a cos 9 - p sin 9 (6)
z' = 9' + a cos 9' + p sin 9' (7)
where a is component of the velocity v/c of the mirror parallel to its surface. Notice that this is the classical aberration effect. A relativistic analysis would lead to a different result.
The measured effect is the difference between z' and z:
z' - z = (9' - 9) + a (cos 9' - cos 9) + p (sin 9' - sin 9) (8)
Taking into account the above equations and making suitable substitutions, one obtains the approximate result:
z' - z = 2aP sin2 z (9)
Replacing a and p by their values in Eqs. (1) and (2),17 one obtains:
z' - z = [(v/c)2 sin2 z].[sin 2(|).sin2D + cos 2(|).sin 2D.cos (d-A) -
- sin 2<|).cos2D.cos2(9-^4)] (10)
Notice that this equation contains a constant term and two periodical components with different periods - one sidereal day [cos (9-^4)] and half a sidereal day [cos2 (9-^4)]. Therefore, from a suitable analysis of the data it should be possible to obtain the speed (v/c), the declination (D) and the right ascension (A) of the motion of the Earth relative to the ether.
Repetition of the Leyden measurements
The Leyden measurements had used four stars close to the North Pole. The difference z-z' was measured in a series of observations, at the times of upper and lower culmination of each star. The observed values of the periodical components of z-z' amounted to less than 1 ", varying from 0.04" for one of the stars to about 0.5" for another. The error of the measurements was estimated as 0.01 ", therefore the effect was regarded as significant. From the Leyden data Courvoisier obtained the results:
A = 104° ± 21°; D = +39° ± 27°; v = 810 ± 215 km/s
17 From this point onward, 9 is used again to represent sidereal time.