Page:Aether and Matter, 1900.djvu/203

place until it is proved to be in definite contradiction, not removable by suitable modification, with another portion of it.

Application to moving Material Media: approximation up to first order

106. We now recall the equations of the free aether, with a view to changing from axes (x, y, z) at rest in the aether to axes (x', y', z') moving with translatory velocity v parallel to the axis of x; so as thereby to be in a position to examine how phenomena are altered when the observer and his apparatus are in uniform motion through the stationary aether. These equations are

${\displaystyle {\begin{array}{ccc}4\pi {\frac {df}{dt}}={\frac {dc}{dy}}-{\frac {db}{dz}}&&-(4\pi c^{2})^{-1}{\frac {da}{dt}}={\frac {dh}{dy}}-{\frac {dg}{dz}}\\\\4\pi {\frac {dg}{dt}}={\frac {da}{dz}}-{\frac {dc}{dx}}&&-(4\pi c^{2})^{-1}{\frac {db}{dt}}={\frac {df}{dz}}-{\frac {dh}{dx}}\\\\4\pi {\frac {dh}{dt}}={\frac {db}{dx}}-{\frac {da}{dy}}&&-(4\pi c^{2})^{-1}{\frac {dh}{dt}}={\frac {dg}{dx}}-{\frac {df}{dy}}.\end{array}}}$

When they are referred to the axes (x', y', z') in uniform motion, so that ${\displaystyle (x',\ y',\ z')=(x-vt,\ y,\ z),\ t'=t}$, then ${\displaystyle d/dx,\ d/dy,\ d/dz}$ become ${\displaystyle d/dx',\ d/dy',\ d/dz'}$, but d/dt becomes ${\displaystyle d/dt'-vd/dx'}$: thus

${\displaystyle {\begin{array}{ccc}4\pi {\frac {df}{dt'}}={\frac {dc'}{dy'}}-{\frac {db'}{dz'}}&&-(4\pi c^{2})^{-1}{\frac {da}{dt'}}={\frac {dh'}{dy'}}-{\frac {dg'}{dz'}}\\\\4\pi {\frac {dg}{dt'}}={\frac {da'}{dz'}}-{\frac {dc'}{dx'}}&&-(4\pi c^{2})^{-1}{\frac {db}{dt'}}={\frac {df'}{dz'}}-{\frac {dh'}{dx'}}\\\\4\pi {\frac {dh}{dt'}}={\frac {db'}{dx'}}-{\frac {da'}{dy'}}&&-(4\pi c^{2})^{-1}{\frac {dh}{dt'}}={\frac {dg'}{dx'}}-{\frac {df'}{dy'}}.\end{array}}}$

where

 ${\displaystyle (a',\ b',\ c')=(a,\ b+4\pi vh,\ c-4\pi vg)}$ ${\displaystyle (f',\ g',\ h')=\left(f,\ g-{\frac {v}{4\pi c^{2}}}c,\ h+{\frac {v}{4\pi c^{2}}}b\right).}$

We can complete the elimination of (f, g, h) and (a, b, c) so