Page:Aether and Matter, 1900.djvu/206

hence bearing in mind that for the system at rest

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

or, what is the same,

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

we have, to the first order,

${\frac {df}{dx}}+{\frac {dg}{dy}}+{\frac {dh}{dz}}={\frac {df'}{dx}}+{\frac {dg'}{dy}}+{\frac {dh'}{dz}}.$

Thus the electrons in the two systems here compared, being situated at the singular points at which the concentration of the electric displacement ceases to vanish, occupy corresponding positions. Again, these electrons are of equal strengths: for, very near an electron, fixed or moving, the values of (f, g, h) and (a, b, c) are practically those due to it, the part due to the remainder of the system being negligible in comparison: also in this correspondence the relation between (f, g, h) and the accented variables is, by § 106

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

hence, since for the single electron at rest (a', b', c') is null, we have, very close to the correlative electron in the moving system, (f, g, h) equal to (f', εg', εh'), where ε, being $\left(1-v^{2}/c^{2}\right)^{-1}$, differs from unity by the second order of small quantities. Thus neglecting the second order, (f, g, h) is equal to (f', g', h') for corresponding points very close to electrons; and, as the amount of electricity inside any boundary is equal to the integral of the normal component of the aethereal displacement taken over the boundary, it follows by taking a very contracted boundary that the strengths of the corresponding electrons in the two systems are the same, to this order of approximation.

108. It is to be observed that the above analytical transformation of the equations applies to any isotropic dielectric medium as well as to free aether: we have only to alter c into