Page:Aether and Matter, 1900.djvu/198

portion of the surrounding aethereal strain, of which the displacement-vector (f, g, h) is the expression, which is associated with the electron and is carried along with the electron in its motion, being as above $-e/4\pi \ .\ (d/dx,\ d/dy,\ d/dz)r^{-1}$. It is to be noticed that the energy of this part of the displacement is closely concentrated around the nucleus of the electron, and not widely diffused as might at first sight appear. The aethereal displacement satisfies the stream-condition

$df/dx+dg/dy+dh/dz=0,\$

except where there are electrons in the effective element of volume: these are analogous to the so-called sources and sinks in the abstract theory of liquid flow, so that when electrons are present the integral of the normal component of the aethereal displacement over the boundary of any region, instead of being null, is equal to the quantity Σe of electrons existing in the region. The other vector which is associated with the aether, namely the magnetic induction (a, b, c), also possesses the stream property; but singular points in its distribution, of the nature of simple poles, do not exist. The motion of an electron involves however a singularity in (a, b, c), of a rotational type, with its nucleus at the moving electron;^[1] and the time-average of this singularity for a very rapid minute steady orbital motion of an electron is analytically equivalent, at distances considerable compared with the dimensions of the orbit, to a magnetic doublet analogous to a source and

1. Namely as the distance r from it diminishes indefinitely, the magnetic induction tends to the form $evr^{-2}\sin {\theta }$, at right angles to the plane of the angle θ between r and the velocity v of the electron: this arises as the disturbance of the medium involved in annulling the electron in its original position and restoring it in the new position to which it has moved. The relations will appear more clearly when visualized by the kinematic representation of Appendix E; or when we pass to the limit in the formulae of Chapter ix relating to the field of a moving charged body of finite dimensions.

   The specification in the text, as a simple pole, only applies for an electron moving with velocity v, when terms of the order $(v/c)^{2}$ are neglected: otherwise the aethereal field close around it is not isotropic and an amended specification derivable from the formulae of Chapter ix must be substituted. In the second-order discussion of Chapter xi this more exact form is implicitly involved, the strength of the electron being determined (§ 111) by the concentration of the aethereal displacement around it. The singularity in the magnetic field which is involved in the motion of the electron, not of course an intrinsic one, has no concentration.