Page:A Dynamical Theory of the Electromagnetic Field.pdf/42

If we determine $\chi$ from the equation

$\nabla ^{2}\chi ={\frac {d^{2}\chi }{dx^{2}}}+{\frac {d^{2}\chi }{dy^{2}}}+{\frac {d^{2}\chi }{dz^{2}}}=J,$

(73)

and F', G', H' from the equations

$F'=F-{\frac {d\chi }{dx}},\ G'=G-{\frac {d\chi }{dy}},\ H'=H-{\frac {d\chi }{dz}},$

(74)

then

${\frac {dF'}{dx}}+{\frac {dG'}{dy}}+{\frac {dH'}{dz}}=0$

(75)

and the equations in (94) become of the form

$k\nabla ^{2}F'=4\pi \mu \left({\frac {a^{2}F'}{dt^{2}}}+{\frac {d}{dxdt}}\left(\Psi +{\frac {d\chi }{dt}}\right)\right)$

(76)

Differentiating the three equations with respect to $x,y$, and $z$, and adding, we find that

$\Psi =-{\frac {d\chi }{dt}}+\varphi (x,y,z)$

(77)

and that

$\left.{\begin{array}{l}k\nabla ^{2}F'=4\pi \mu {\frac {d^{2}F}{dt^{2}}},\\\\k\nabla ^{2}G'=4\pi \mu {\frac {d^{2}G'}{dt^{2}}},\\\\k\nabla ^{2}H'=4\pi \mu {\frac {d^{2}H'}{dt^{2}}}.\end{array}}\right\}$

(78)

Hence the disturbances indicated by F', G', H' are propagated with the velocity $V={\sqrt {\tfrac {k}{4\pi \mu }}}$ through the field: and since

${\frac {dF'}{dx}}+{\frac {dG'}{dy}}+{\frac {dH'}{dz}}=0$

the resultant of these disturbances is in the plane of the wave.

(99) The remaining part of the total disturbances F, G, H being the part depending on $\chi$ is subject to no condition except that expressed in the equation

${\frac {d\Psi }{dt}}+{\frac {d^{2}\chi }{dt^{2}}}=0$

If we perform the operation $\nabla ^{2}$ on this equation, it becomes

$ke={\frac {dJ}{dt}}-k\nabla ^{2}\varphi (x,y,z)$

(79)

Since the medium is a perfect insulator, $e$, the free electricity, is immoveable, and therefore ${\tfrac {dJ}{dt}}$ is a function of $x,y,z$, and the value of J is either constant or zero, or uniformly increasing or diminishing with the time; so that no disturbance depending on J can be propagated as a wave.

(100) The equations of the electromagnetic field, deduced from purely experimental evidence, show that transversal vibrations only can be propagated. If we were to go beyond our experimental knowledge and to assign a definite density to a substance which