Page:Aether and Matter, 1900.djvu/209

CHAPTER XI:
MOVING MATERIAL SYSTEM:

Approximation carried to the second order

110. THE results above obtained have been derived from the correlation developed in § 106, up to the first order of the small quantity v/c, between the equations for aethereal vectors here represented by (f', g', h') and (a', b', c') referred to the axes (x', y', z') at rest in the aether and a time t", and those for related aethereal vectors represented by (f, g, h) and (a, b, c) referred to axes (x', y', z') in uniform translatory motion and a time t'. But we can proceed further, and by aid of a more complete transformation institute a correspondence which will be correct to the second order. Writing as before t" for $t'-{\frac {v}{c^{2}}}\epsilon x'$, the exact equations for (f, g, h) and (a, b, c) referred to the moving axes (x', y', z') and time t' are, as above shown, equivalent to

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

Now write

$\left(x_{1},\ y_{1},\ z_{1}\right)$ for $\left(\epsilon ^{\frac {1}{2}}x',\ y',\ z'\right)$

$\left(a_{1},\ b_{1},\ c_{1}\right)$ for $\left(\epsilon ^{-{\frac {1}{2}}}a',\ b',\ c'\right)$ or $\left(\epsilon ^{-{\frac {1}{2}}}a,\ b+4\pi vh,\ c-4\pi vg\right)$