Page:Aether and Matter, 1900.djvu/212

argument still applies if the surface be taken to surround the electron under consideration very closely, because then the wholly preponderating part of each vector is that which belongs to the adjacent electron.^[1]

112. We require however to construct a correlative system devoid of the translatory motion in which the strengths of the electrons shall be equal instead of proportional, since motion of a material system containing electrons cannot alter their strengths. The principle of dynamical similarity will effect this.

We have in fact to reduce the scale of the electric charges, and therefore of ${\frac {df}{dx}}+{\frac {dg}{dy}}+{\frac {dh}{dz}}$, in a system at rest in the ratio $\epsilon ^{-{\frac {1}{2}}}$. Apply therefore a transformation

$(x,\ y,\ z)=k\left(x_{1},\ y_{1},\ z_{1}\right),\ t=lt_{1},$ $(a,b,c)=\vartheta (a_{1},\ b_{1},\ c_{1}),\ (f,\ g,\ h)=\epsilon ^{-{\frac {1}{2}}}k\left(f_{1},\ g_{1},\ h_{1}\right);$

and the form of the fundamental circuital aethereal relations will not be changed provided $k=l$ and $\vartheta =\epsilon ^{-{\frac {1}{2}}}k$. Thus we may have k and l both unity and $\vartheta =\epsilon ^{-{\frac {1}{2}}}$; so that no further change of scale in space and time is required, but only a diminution of (a, b, c) in the ratio $\epsilon ^{-{\frac {1}{2}}}$.

We derive the result, correct to the second order, that if the internal forces of a material system arise wholly from electrodynamic actions between the systems of electrons which constitute the atoms, then an effect of imparting to a steady material system a uniform velocity of translation is to produce a uniform contraction of the system in the direction of the motion, of amount $\epsilon ^{-{\frac {1}{2}}}$ or $1-{\frac {1}{2}}v^{2}/c^{2}$. The electrons will occupy corresponding positions in this contracted system, but the aethereal displacements in the space around them will not correspond: if (f, g, h) and (a, b, c) are those of the moving system, then the electric and magnetic displacements at corresponding points of the fixed systems will be the values that the vectors

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

1. This result follows more immediately from § 110, which shows that corresponding densities of electrification are equal, while corresponding volumes are as $\epsilon ^{\frac {1}{2}}$ to unity.