Page:Aether and Matter, 1900.djvu/205

107. It is to be observed that this factor ε only differs from unity by $(v/c)^{2}$, which is of the second order of small quantities; hence we have the following correspondence when that order is neglected. Consider any aethereal system, and let the sequence of its spontaneous changes referred to axes (x', y', z') moving uniformly through the aether with velocity (v, 0, 0) be represented by values of the vectors (f, g, h) and (a, b, c) expressed as functions of x', y', z' and t', the latter being the time measured in the ordinary manner: then there exists a correlated aethereal system whose sequence of spontaneous changes referred to axes (x', y', z') at rest are such that its electric and magnetic vectors (f', g' , h') and (a', b', c') are functions of the variables x', y', z' and a time-variable t", equal to $t'-{\frac {v}{c^{2}}}x'$, which are the same as represent the quantities

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

and

$(a,\ b+4\pi vh,\ c-4\pi vg)$

belonging to the related moving system when expressed as functions of the variables x', y', z' and t'.

Conversely, taking any aethereal system at rest in the aether, let the sequence of its changes be represented by (f', g', h') and (a', b', c') expressed as functions of the coordinates (x, y, z) and of the time t'. In these functions change t' into $t-{\frac {v}{c^{2}}}x$: then the resulting expressions are the values of

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

and

$(a,\ b+4\pi vh,\ c-4\pi vg),$

for a system in uniform motion through the aether, referred to axes (x, y, z) moving along with it, and to the time t. In comparing the states of the two systems, we have to the first order

${\frac {df}{dx}}$	equal to	${\frac {df'}{dx}}-{\frac {v}{c'}}{\frac {df'}{dt}}$
${\frac {d}{dy}}\left(g-{\frac {v}{4\pi c^{2}}}c\right)$	„	${\frac {dg'}{dy}}$
${\frac {d}{dz}}\left(h+{\frac {v}{4\pi c^{2}}}b\right)$	„	${\frac {dh'}{dz}};$