Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/230

where $t$ denotes the time, $p$ the period, and $a_{1},a_{2}$ functions of the coordinates. It follows that

$[{\ddot {\xi }}]_{\text{Ave}}=-{\frac {4\pi ^{2}}{p^{2}}}[\xi ]_{\text{Ave}},$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$ (2)
etc.,

4. Now, on the electrical theory, these motions are excited by electrical forces, which are of two kinds, distinguished as electrostatic and electrodynamic. The electrostatic force is determined by the electrostatic potential. If we write $q$ for the actual value of the potential, and $[q]_{\text{Ave}}$ for its value as averaged in the manner specified above, the components of the actual electrostatic force will be

$-{\frac {dq}{dx}},$$-{\frac {dq}{dy}},$$-{\frac {dq}{dz}};$

and for the average values of these components in the small spaces described above we may write

$-{\frac {d[q]_{\text{Ave}}}{dx}},$$-{\frac {d[q]_{\text{Ave}}}{dy}},$$-{\frac {d[q]_{\text{Ave}}}{dz}},$

for it will make no difference whether we take the average before or after differentiation.

5. The electrodynamic force is determined by the acceleration of electrical flux in all parts of the field, but physicists are not entirely agreed in regard to the laws by which it is determined. This difference of opinion is however of less importance, since it will not affect the result if electrical fiuxes are always solenoidal. According to the most simple law, the components of the force are given by the volume-integrals

$-\iiint {\frac {\ddot {\xi }}{r}}dv,$$-\iiint {\frac {\ddot {\eta }}{r}}dv,$$-\iiint {\frac {\ddot {\zeta }}{r}}dv,$

where $dv$ represents an element of volume, and $r$ the distance of this element from the point for which the value of the electromotive force is to be determined. In other words, the components of the force at any point are determined from the components of acceleration in all parts of the field by the same process by which (in the theories of gravitation, etc.) the value of the potential at any point is determined from the density of matter in all parts of space, except that the sign is to be reversed. Adopting this law, provisionally at leasts we may express it by saying that the components of electrodynamic force are equal to the potentials taken negatively of the components of acceleration of electrical flux. And we may write, for brevity,

$-{\text{Pot }}{\ddot {\xi }},$$-{\text{Pot }}{\ddot {\eta }},$$-{\text{Pot }}{\ddot {\zeta }},$

for the components of force, using the symbol $Pot$ to denote the operation by which the potential of a mass is derived from its