Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/456

analytical theory of electrons, nothing more would be required than to modify the formulae by writing e (the charge of an electron) in place of ρdxdydz. That this is not the case was shown^[1] a few years after the publication of the Versuch.

Consider, for example, the formula for the scalar potential at any point in the aether,

$\phi =\textstyle \iiint ({\bar {\rho }}^{\prime }/r)\ dx^{\prime }\ dy^{\prime }\ dz^{\prime }$,

where the bar indicates that the quantity underneath it is to have its retarded value.^[2]

This integral, in which the integration is extended over all elements of space, must be transformed before the integration can be taken to extend over moving elements of charge. Let de′ denote the sum of the electric charges which are accounted for under the heading of the volume-element dx′dy′dz′ in the above integral. This quantity de′ is not identical with ${\bar {\rho }}^{\prime }dx^{\prime }dy^{\prime }dz^{\prime }$. For, to take the simplest case, suppose that it is required to compute the value of the potential-function for the origin at the time t, and that the charge is receding from the origin along the axis of x with velocity u. The charge which is to be ascribed to any position x is the charge which occupies that position at the instant t - x/c; so that when the reckoning is made according to intervals of space, it is necessary to reckon within a segment (x₂ – x₁) not the electricity which at any one instant occupies that segment, but the electricity which at the instant (t - x₃/c) occupies a segment (x₂ – x′₁), where x′₁ denotes the point from which the electricity streams to x₁, in the interval between the instants (t - x₂/c) and (t - x₁/c). We have evidently

$x_{1}-x_{1}^{\prime }=u(x_{2}-x_{1})/c,\qquad {\text{ or }}\qquad x_{2}-x_{1}^{\prime }=(x_{2}-x_{1})(1+u/c)$.

For this case we should therefore have

$de^{\prime }={\frac {x_{2}-x_{1}^{\prime }}{x_{2}-x_{1}}}{\bar {\rho }}^{\prime }\ dx^{\prime }dy^{\prime }dz^{\prime }=\left(1+{\frac {u}{c}}\right){\bar {\rho }}^{\prime }\ dx^{\prime }dy^{\prime }dz^{\prime }$.

1. E. Wiechert, Arch. Néerl. (2) v (1900), p. 549. Cf. also A. Liénard, L' Éclairage élect. xvi (1898), pp. 5, 53, 106.
2. Cf. p. 298.