Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/66

Whenever, therefore, $\omega$ is discontinuous at surfaces, the expressions ${\text{Pot }}\nabla .\omega$ and ${\text{New }}\nabla .\omega$ must be regarded as implicitly including the surface-integrals

$\iint {\frac {1}{[\rho '-\rho ]_{0}}}\Delta \omega '.d\sigma '$⁠and⁠$\iint {\frac {\rho '-\rho }{[\rho '-\rho ]_{0}^{3}}}\Delta \omega '.d\sigma '$

respectively, relating to such surfaces, and the expressions ${\text{Pot }}\nabla \times \omega$ and ${\text{Lap }}\nabla \times \omega$ as including the surface-integrals

$\iint {\frac {1}{[\rho '-\rho ]_{0}}}d\sigma '\times \Delta \omega '$⁠and⁠$\iint {\frac {\rho '-\rho }{[\rho '-\rho ]_{0}^{3}}}\times [d\sigma '\times \Delta \omega ']$

respectively, relating to such surfaces.

101. We have already seen that if $\omega$ is the curl of any vector function of position, $\nabla .\omega =0.$ (No. 68.) The converse is evidently true, whenever the equation $\nabla .\omega =0$ holds throughout all space, and $\omega$ has in general a definite potential; for then

$\omega =\nabla \times {\frac {1}{4\pi }}{\text{Lap }}\omega .$

Again, if $\nabla .\omega =0$ within any aperiphractic space ${\text{A}},$ contained within finite boundaries, we may suppose that space to be enclosed by a shell ${\text{B}}$ having its inner surface coincident with the surface of ${\text{A}}.$ We may imagine a function of position $\omega ',$ such that $\omega '=\omega$ in ${\text{A}},\omega '=0$ outside of the shell ${\text{B}}'$ and the integral $\iiint \omega '.\omega 'dv$ for ${\text{B}}$ has the least value consistent with the conditions that the normal component of $\omega '$ at the outer surface is zero, and at the inner surface is equal to that of $\omega ,$ and that in the shell $\nabla .\omega '=0$ (compare No. 90). Then $\nabla .\omega '=0$ throughout all space, and the potential of $\omega '$ will have in general a definite value. Hence,

$\omega '=\nabla \times {\frac {1}{4\pi }}{\text{Lap }}\omega ',$

and $\omega$ will have the same value within the space ${\text{A}}.$

^[1]102. Def.—If $\omega$ is a vector function of position in space, the Maxwellian^[2] of $\omega$ is a scalar function of position, defined by the equation

${\text{Max }}\omega =\iiint {\frac {\rho '-\rho }{[\rho '-\rho ]_{0}^{3}}}.\omega '\,dv'.$

(Compare No. 92.) From this definition the following properties are easily derived. It is supposed that the functions $\omega$ and $u$ are such that their potentials have in general definite values.

${\begin{aligned}{\text{Max }}\omega &=\nabla .{\text{Pot }}\omega ={\text{Pot }}\nabla .\omega ,\\\nabla {\text{Max }}\omega &=\nabla \nabla .{\text{Pot }}\omega ={\text{New }}\nabla .\omega ,\\{\text{Max }}\nabla u&=-4\pi u,\\4\pi u&=\nabla \times {\text{Lap }}\omega -\nabla {\text{Max }}\omega .\end{aligned}}$

1. [The foregoing portion of this paper was printed in 1881, the rest in 1884.]
2. The frequent occurrence of the integral in Maxwell's Treatise on Electricity and Magnetism has suggested this name.