Page:Aether and Matter, 1900.djvu/215

$\left({\frac {\epsilon ^{-1}\cos \theta '}{V'}}-{\frac {v}{c^{2}}}\right)^{2}+{\frac {\epsilon ^{-1}\cos ^{2}\theta '}{V'^{2}}}={\frac {1}{V^{2}}},$

so that neglecting $(v/c)^{3}$,

$V'=V-{\frac {v}{\mu ^{2}}}\cos \theta '-{\frac {1}{2}}\left(1-\mu ^{-2}\right){\frac {v^{2}}{\mu c}}\left(1+3\cos ^{2}\theta '\right),$

where $\mu =c/V$, of which the last term is the general form of the second order correction to Fresnel's expression. In free aether, for which μ, is unity, this formula represents the velocity relative to the moving axes of an unaltered wave-train, as it ought to do.

As (f, g, h) and (a, b, c) are in the same phase in the free transparent aether, when one of them is null so is the other: hence in any experimental arrangement, regions where there is no disturbance in the one system correspond to regions where there is no disturbance in the other. As optical measurements are usually made by the null method of adjusting the apparatus so that the disturbance vanishes, this result carries the general absence of effect of the Earth's motion in optical experiments, up to the second order of small quantities.

Influence of translatory motion on the Structure of a Molecule: the law of Conservation of Mass

114. As a simple illustration of the general molecular theory, let us consider the group formed of a pair of electrons of opposite signs describing steady circular orbits round each other in a position of rest:^[1] we can assert from the correlation, that when this pair is moving through the aether with velocity v in a direction lying in the plane of their orbits, these orbits relative to the translatory motion will be flattened along the direction of v to ellipticity $1-{\frac {1}{2}}v^{2}/c^{2}$, while there will be a first-order retardation of phase in each orbital motion when the electron is in front of the mean position combined with acceleration when behind it so that on the whole the period will be changed only in the second-order ratio $1+{\frac {1}{2}}v^{2}/c^{2}$. The specification of the orbital modification produced by the

1. The orbital velocities are in this illustration supposed so small that radiation is not important. Cf. §§ 151 — 6 infra.