Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/67

If the values of ${\text{Lap Lap }}\omega ,{\text{New Max }}\omega ,$ and ${\text{Max New }}u$ are in general definite, we may add

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

In other words: The Maxwellian is the divergence pf the potential, $-{\frac {\text{Max}}{4\pi }}$ and $\nabla$ are inverse operators for scalars and irrotational vectors, for vectors in general $-{\frac {1}{4\pi }}\nabla {\text{Max}}$ is an operator which separates the irrotational from the solenoidal part. For scalars and irrotational vectors, ${\frac {-1}{4\pi }}{\text{Max New}}$ and ${\frac {-1}{4\pi }}{\text{New Max}}$ give the potential, for solenoidal vectors ${\frac {1}{4\pi }}{\text{Lap Lap}}$ gives the potential, for vectors in general ${\frac {-1}{4\pi }}{\text{New Max}}$ gives the potential of the irrotational part, and ${\frac {1}{4\pi }}{\text{Lap Lap}}$ the potential of the solenoidal part.

103. Def.—The following double volume-integrals are of frequent occurrence in physical problems. They are all scalar quantities, and none of them functions of position in space, as are the single volume-integrals which we have been considering. The integrations extend over all space, or as far as the expression to be integrated has values other than zero.

The mutual potential, or potential product, of two scalar functions of position in space is defined by the equation

${\text{Pot}}(u,w)=\iiint \iiint {\frac {uw'}{r}}dv\,dv'=\iiint u{\text{Pot }}w\,dv=\iiint w{\text{Pot }}u\,dv.$

In the double volume-integral, $r$ is the distance between the two elements of volume, and $u$ relates to $dv$ as $w'$ to $dv'.$

The mutual potential, or potential product, of two vector functions of position in space is defined by the equation

${\text{Pot}}(\phi ,\omega )=\iiint \iiint {\frac {\phi .\omega '}{r}}dv\,dv'=\iiint \phi .{\text{Pot }}w\,dv=\iiint \omega .{\text{Pot }}\phi \,dv.$

The mutual Laplacian, or Laplacian product, of two vector functions of position in space is defined by the equation

${\text{Lap}}(\phi ,\omega )=\iiint \iiint \omega .{\frac {\rho '-\rho }{r^{3}}}\times \phi 'dv\,dv'=\iiint \omega .{\text{Lap }}\phi \,dv=\iiint \phi .{\text{Lap }}\omega \,dv.$

The Newtonian product of a scalar and a vector function of position in space is defined by the equation

${\text{New}}(u,\omega )=\iiint \iiint \omega .{\frac {\rho '-\rho }{r^{3}}}u'\,dv\,dv'=\iiint \omega .{\text{New }}u\,dv.$