Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/381

is approximately a plane wave, the opposite sides of the ring representing the two phases of the wave. When one of these rings has become detached from the radiator, the energy contained may subsequently be regarded as travelling outwards with it.

To discuss the problem analytically^[1] we take the axis of the radiator as axis of z, and the centre of the spark-gap as origin. The field may be regarded as due to an electric doublet formed of a positive and an equal negative charge, displaced from each other along the axis of the vibrator, and of moment

$Ae^{-p_{1}t}\sin(2\pi ct/\lambda )$,

the factor $e^{-p_{1}t}$ being inserted to represent the damping.

The simplest method of proceeding, which was suggested by FitzGerald,^[2] is to form the retarded potentials φ and a of L. Lorenz.^[3] These are determined in terms of the charges and their velocities by the equations

$\phi =\textstyle \sum {\frac {(e)_{t-r/c}}{r}},\qquad a_{z}=\textstyle \sum {\frac {(e{\dot {z}})_{t-r/c}}{r}}$,

whence it is readily shown that in the present case

$\phi =-\partial F/\partial z,\qquad \mathbf {a} =(0,0,\partial F/\partial t)$,

where

$F={\frac {Ae^{-p_{1}(t-r/c)}}{r}}\sin {\frac {2\pi }{\lambda }}(ct-r)$.

The electric and magnetic forces are then determined by the equations

$\mathbf {E} =c^{2}{\text{ curl }}\phi -\mathbf {\dot {a}} ,\qquad \mathbf {H} ={\text{curl }}\mathbf {a}$.

It is found that the electric force may be regarded as compounded of a force φ₂, parallel to the axis of the vibrator and depending at any instant only on the distance from the vibrator, together with a force φ₁ sin θ acting in the meridian plane

1. Cf. Karl Pearson and A. Lee, Phil, Trans, cxciii (1899), p. 165.
2. Brit. Assoc. Rep., Leeds (1890), p. 755.
3. Cf. p. 298. The use of retarded potentials was also recommended in the following year by Poincaré, Comptes Rendus, cxiii (1891), p. 515.