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 (the charge of an electron) in place of. That this is not the case was shown a few years after the publication of the Versuch.

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

where the bar indicates that the quantity underneath it is to have its retarded value.

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 denote the sum of the electric charges which are accounted for under the heading of the volume-element  in the above integral. This quantity 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, and that the charge is receding from the origin along the axis of with velocity. The charge which is to be ascribed to any position is the charge which occupies that position at the instant ; so that when the reckoning is made according to intervals of space, it is necessary to reckon within a segment  not the electricity which at any one instant occupies that segment, but the electricity which at the instant  occupies a segment, where  denotes the point from which the electricity streams to , in the interval between the instants  and. We have evidently

For this case we should therefore have