Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/318

appreciable in ordinary laboratory experiments, would be capable of accounting for the propagation of electrical effects through space with a finite velocity. We have seen that in Neumann's theory the electric force E was determined by the equation

$\mathbf {E} =c^{2}\mathrm {grad} \ \phi -{\overset {.}{\mathbf {a} }}$,      (1)

where φ denotes the electrostatic potential defined by the equation

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

ρ′ being the density of electric charge at the point (x′, y′, z′), and where a denotes the vector-potential, defined by the equation

$\mathbf {a} =\iiint (\mathbf {\iota ^{\prime }} /r)dx^{\prime }\ dy^{\prime }\ dz^{\prime }$,

ι′ being the conduction-current at (x′, y′, z′). We suppose the specific inductive capacity and the magnetic permeability to be everywhere unity.

Lorenz proposed to replace these by the equations

$\mathbf {a} =\iiint \{\mathbf {\iota ^{\prime }} (t-r/c)/r\}dx^{\prime }\ dy^{\prime }\ dz^{\prime }$,

$\mathbf {a} =\iiint \{\mathbf {\iota ^{\prime }} (t-r/c)/r\}dx^{\prime }\ dy^{\prime }\ dz^{\prime }$;

the change consists in replacing the values which ρ′ and ι′ have at the instant t by those which they have at the instant (t - r/c), which is the instant at which a disturbance travelling with velocity c must leave the place (x′, y′, z′) in order to arrive at the place (x, y, z) at the instant t. Thus the values of the potentials at (x, y, z) at any instant t would, according to Lorenz's theory, depend on the electric state at the point (x′, y′, z′) at the previous instant (t - r/c): as if the potentials were propagated outwards from the charges and currents with velocity c. The functions φ and a formed in this way are generally known as the retarded potentials.