Page:Aether and Matter, 1900.djvu/204

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that only the vectors denoted by accented symbols shall remain, by substituting from these latter formulae: thus

g=g'+{\frac {v}{4\pi c^{2}}}(c'+4\pi vg),

so that

\epsilon ^{-1}g=g'+{\frac {v}{4\pi c^{2}}}c',

where ε is equal to $\left(1-v^{2}/c^{2}\right)^{-1}$ , and exceeds unity;

and

b=b'-4\pi v\left(h'-{\frac {v}{4\pi c^{2}}}b\right)

so that

\epsilon ^{-1}b=b'-4\pi vh';

giving the general relations

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

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

Hence

4\pi {\frac {df'}{dt'}}={\frac {dc'}{dy'}}-{\frac {db'}{dz'}}

$4\pi \epsilon {\frac {dg'}{dt'}}={\frac {da'}{dz'}}-\left({\frac {d}{dx'}}+{\frac {v}{c^{2}}}\epsilon {\frac {d}{dt'}}\right)c'$

$4\pi \epsilon {\frac {dh'}{dt'}}=\left({\frac {d}{dx'}}+{\frac {v}{c^{2}}}\epsilon {\frac {d}{dt'}}\right)b'-{\frac {da'}{dy'}}$

$-\left(4\pi c^{2}\right)^{-1}{\frac {da'}{dt'}}={\frac {dh'}{dy'}}-{\frac {dg'}{dz'}}$

$-\left(4\pi c^{2}\right)^{-1}\epsilon {\frac {db'}{dt'}}={\frac {df'}{dz'}}-\left({\frac {d}{dx'}}+{\frac {v}{c^{2}}}\epsilon {\frac {d}{dt'}}\right)h'$

$-\left(4\pi c^{2}\right)^{-1}\epsilon {\frac {dc'}{dt'}}=\left({\frac {d}{dx'}}+{\frac {v}{c^{2}}}\epsilon {\frac {d}{dt'}}\right)g'-{\frac {df'}{dy'}}.$

Now change the time-variable from t' to t", equal to $t'-{\frac {v}{c^{2}}}\epsilon x'$ ; this will involve that ${\frac {d}{dx'}}+{\frac {v}{c^{2}}}\epsilon {\frac {d}{dt'}}$ is replaced by ${\frac {d}{dx'}}$ , while the other differential operators remain unmodified; thus the scheme of equations reverts to the same type as when it was referred to axes at rest, except as regards the factors ε on the left-hand sides.