Page:Aether and Matter, 1900.djvu/210

$\left(f_{1},\ g_{1},\ h_{1}\right)$	for	$\left(\epsilon ^{-{\frac {1}{2}}}f',\ g',\ h'\right)$	or	$\left(\epsilon ^{-{\frac {1}{2}}}f,\ g-{\frac {v}{4\pi c^{2}}}c,\ h+{\frac {v}{4\pi c^{2}}}b\right)$
$dt_{1}$	for	$\epsilon ^{-{\frac {1}{2}}}dt''$	or	$\epsilon ^{-{\frac {1}{2}}}\left(dt'-{\frac {v}{c^{2}}}\epsilon dx'\right),$

where $\epsilon =\left(1-v^{2}/c^{2}\right)^{-1}$; and it will be seen that the factor ε is absorbed, so that the scheme of equations, referred to moving axes, which connects together the new variables with subscripts, is identical in form with the Maxwellian scheme of relations for the aethereal vectors referred to fixed axes. This transformation, from (x', y', z') to $(x_{1},\ y_{1},\ z_{1})$ as dependent variables, signifies an elongation of the space of the problem in the ratio $\epsilon ^{\frac {1}{2}}$ along the direction of the motion of the axes of coordinates. Thus if the values of $\left(f_{1},\ g_{1},\ h_{1}\right)$ and $(a_{1},\ b_{1},\ c_{1})$ given as functions of $x_{1},\ y_{1},\ z_{1},\ t_{1}$ express the course of spontaneous change of the aethereal vectors of a system of moving electrons referred to axes $(x_{1},\ y_{1},\ z_{1})$ at rest in the aether, then

$\left(\epsilon ^{-{\frac {1}{2}}}f,\ g-{\frac {v}{4\pi c^{2}}}c,\ h+{\frac {v}{4\pi c^{2}}}b\right)$

and

$\left(\epsilon ^{-{\frac {1}{2}}}a,\ b+4\pi vh,\ c-4\pi vg\right),$

expressed by the same functions of the variables

$\epsilon ^{\frac {1}{2}}x',\ y',\ z',\ \epsilon ^{-{\frac {1}{2}}}t'-{\frac {v}{c^{2}}}\epsilon ^{\frac {1}{2}}x',$

will represent the course of change of the aethereal vectors (f, g, h) and (a, b, c) of a correlated system of moving electrons referred to axes of (x', y', z') moving through the aether with uniform translatory velocity (v, 0, 0). In this correlation between the courses of change of the two systems, we have

${\frac {d\left(\epsilon ^{-{\frac {1}{2}}}f\right)}{d\left(\epsilon ^{\frac {1}{2}}x'\right)}}$	equal to	${\frac {df_{1}}{dx_{1}}}-{\frac {v}{c^{2}}}{\frac {df_{1}}{dt_{1}}},$
${\frac {d}{dy'}}\left(g-{\frac {v}{4\pi c^{2}}}c\right)$	„	${\frac {dg_{1}}{dy_{1}}}$
${\frac {d}{dz}}\left(h+{\frac {v}{4\pi c^{2}}}b\right)$	„	${\frac {dh_{1}}{dz_{1}}},$

where

${\frac {dc}{dy'}}-{\frac {db}{dz'}}=4\pi \left({\frac {df}{dt'}}-v{\frac {df}{dx'}}\right)$