Page:SearleEllipsoid.djvu/2

When any system of electric charges moves with uniform velocity through the aether, the electromagnetic field, when referred to axes moving forwards with the charges, can be completely defined by means of a quantity $\mathbf {\Psi }$, as was first shown by Prof. J. J. Thomson.^[1] The electric force $\mathbf {E}$ and the magnetic force $\mathbf {H}$ are simple functions of $\mathbf {\Psi }$. But besides $\mathbf {E}$ and $\mathbf {H}$ there is another vector of great importance, viz. the mechanical force $\mathbf {F}$ experienced by a unit charge moving with the rest of the system. The value of $\mathbf {F}$ I have shown {§ 10} to be given by the vector equation

$\mathbf {F} =\mathbf {E} +\mu V\mathbf {uH}$.

(1)

The equations of the field are {§ 4}

$\mathrm {curl} \ \mathbf {F} =0$,

(2)

$\mathbf {H} =\mathrm {K} V\mathbf {uE}$.

(3)

If $v={\tfrac {1}{\sqrt {\mathrm {K} \mu }}}$ is the velocity of light, and if $\alpha$ stand for $1-{\tfrac {u^{2}}{v^{2}}}$, then when the motion takes place parallel to the axis of $x$, we have {§ 4}

$\mathbf {F} _{1}=-{\frac {d\mathbf {\Psi } }{dx}},\ \mathbf {F} _{2}=-{\frac {d\mathbf {\Psi } }{dy}},\ \mathbf {F} _{3}=-{\frac {d\mathbf {\Psi } }{dz}}$,

(4)

$\mathbf {E} _{1}=-{\frac {d\mathbf {\Psi } }{dx}},\ \mathbf {E} _{2}=-{\frac {1}{\alpha }}{\frac {d\mathbf {\Psi } }{dy}},\ \mathbf {E} _{3}=-{\frac {1}{\alpha }}{\frac {d\mathbf {\Psi } }{dz}}$,

(5)

$\mathbf {H} _{1}=0,\ \mathbf {H} _{2}={\frac {\mathrm {K} u}{\alpha }}{\frac {d\mathbf {\Psi } }{dz}},\ \mathbf {H} _{3}=-{\frac {\mathrm {K} u}{\alpha }}{\frac {d\mathbf {\Psi } }{dy}}$,

(6)

From these equations, since $\mathbf {E}$ has no divergence,

$\alpha {\frac {d^{2}\mathbf {\Psi } }{dx^{2}}}+{\frac {d^{2}\mathbf {\Psi } }{dy^{2}}}+{\frac {d^{2}\mathbf {\Psi } }{dt^{2}}}=0$.

(7)

Here, and throughout the paper, the axes are supposed to move forward with the same velocity as the electrical charges.

Prof. W. B. Morton has considered the motion of an ellipsoid in a paper read before the Physical Society on 27th March, 1896.^[2] He obtains the two following results, viz. : (1) that the distribution of electricity is the same as if the ellipsoid is at rest, and (2) the value of $\mathbf {\Psi }$ when the ellipsoid moves along one of its axes.

Prof. Morton obtains his result by the assumption first

1. Phil. Mag. July 1889.
2. Proc. Phys. Soc. No. 71, August 1896, p. 180; Phil. Mag. xli. p. 488.