Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/457

In the general case, it is only necessary to replace u by the component of velocity of the electric charge in the direction of the radius vector from the point at which the potential is to be computed. This component may be written $v\cos({\hat {v.r}})$, where r is measured positively from the point in question to the charge, and v denotes the velocity of the charge. Thus

$cde^{\prime }=\{c+v\cos({\hat {v.r}})\}{\bar {\rho }}^{\prime }\ dx^{\prime }\ dy^{\prime }\ dz^{\prime }$,

and therefore

$\phi =c\int {\frac {de^{\prime }}{c{\bar {r}}+(\mathbf {{\bar {r}}.{\bar {v}}} )}}$,

where the integration is extended over all the charges in the field, and the bars over the letters imply that the position of the charge considered is that which it occupied at the instant $t-{\bar {r}}/c$. In the same way the vector-potential may be shown to have the value

$\mathbf {a} =c\int {\frac {\mathbf {\bar {v}} de^{\prime }}{c{\bar {r}}+(\mathbf {{\bar {r}}.{\bar {v}}} )}}$.

Meanwhile the unsettled problem of the relative motion of earth and aether was provoking a fresh series of experimental investigations. The most interesting of these was due to FitzGerald,^[1] who shortly before his death in February, 1901, commenced to examine the phenomena manifested by a charged electrical condenser, as it is carried through space in consequence of the terrestrial motion. On the assumption that a moving charge develops a magnetic field, there will be associated with the condenser a magnetic force at right angles to the lines of electric force and to the direction of the motion: magnetic energy must therefore be stored in the medium, when the plane of the condenser includes the direction of the drift; but when the plane of the condenser is at right angles to the terrestrial motion, the effects of the opposite charges neutralize each other. FitzGerald's original idea was that, in order to supply the magnetic energy, there must be a mechanical drag on the condenser at the moment of

1. FitzGerald's Scientific Writings, p. 557.