Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/272

units of electricity, or the reciprocal of the velocity of light. For this satisfies the general equation

$-\nabla ^{2}{\mathfrak {D}}=c^{2}d^{2}{\mathfrak {D}}/dt^{2},$

(5)

as well as the so-called "equation of continuity," and also satisfies the special conditions that when $t<0$

$4\pi {\mathfrak {D}}=m_{0}(3r^{-5}x\rho -r^{-3i})$

outside of the unit sphere, and that at any time at the surface of this sphere

$4\pi {\mathfrak {D}}=m(3x\rho -i),$

if we consider the terms containing the factor $c$ as negligible, when not compensated by large values of $r.$ That equation (4) satisfies the general conditions is easily verified, if we set

$u=r^{-1}{\text{F}}(t-cr),$

(6)

and observe that

$-\nabla ^{2}u=c^{2}d^{2}u/dt^{2},$

(7)

and that the three components of ${\mathfrak {D}}$ are given by the equations

$4\pi f$	$=-d^{2}u/dy^{2}-d^{2}u/dz^{2}$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ (8)
$4\pi g$	$=d^{2}u/dxdy$
$4\pi h$	$=d^{2}u/dxdz$

Equation (4) shows that the changes of the electrical displacement are represented by three systems of spherical waves, of forms determined by the rapidity of the discharge of the system (A, B), which expand with the velocity of light with amplitudes diminishing as $r^{-3},r^{-2},$ and $r^{-1},$ respectively. Outside of these waves, the electrical displacement is unchanged, inside of them it is zero.

If we write (with Maxwell) $-d{\mathfrak {A}}/dt$ for the force of electrodynamic induction at any point, and suppose its rectangular components calculated from those of $-d^{2}{\mathfrak {D}}/dt^{2}$ by the formula used in calculating the potential of a mass from its density, we shall have by Poisson's theorem

$\nabla ^{2}(d{\mathfrak {A}}/dt)=4\pi d^{2}{\mathfrak {D}}/dt^{2},$

or by (5),

$\nabla ^{2}(d{\mathfrak {A}}/dt)=-4\pi c^{-2}\nabla ^{2}{\mathfrak {D}},$

whence

$d{\mathfrak {A}}/dt=-4\pi c^{-2}{\mathfrak {D}}.$

(9)

From this, with (4), and the general equation

$d{\mathfrak {A}}/dt+4\pi c^{-2}{\mathfrak {D}}+\nabla {\text{V}}=0,$

we see that during the discharge of the system (A, B) the electrostatic force $-\nabla {\text{V}}$ vanishes throughout all space, while its place is taken by a precisely equal electrodynamic force $-d{\mathfrak {A}}/dt.$ This electrodynamic force remains unchanged at every point until the passage of the waves, after which the electrostatic force, the electrodynamic force, and the displacement, have the permanent value zero.