Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/61

92. If $u$ or $\omega$ is a scalar or vector function of position in space, we may write ${\text{Pot }}u,{\text{New }}u,{\text{Pot }}\omega ,{\text{Lap }}\omega$ for the volume-integrals of ${\text{pot }}u',$ etc, taken as functions of $\rho ';$ i.e., we may set

${\text{Pot }}u$	$=\iiint {\text{pot }}u'dv'$	$=\iiint {\frac {u'}{[\rho '-\rho ]_{0}}}dv',$
${\text{New }}u$	$=\iiint {\text{new }}u'dv'$	$=\iiint {\frac {\rho '-\rho }{[\rho '-\rho ]_{0}^{3}}}u'\,dv',$
${\text{Pot }}\omega$	$=\iiint {\text{pot }}\omega 'dv'$	$=\iiint {\frac {\omega '}{[\rho '-\rho ]_{0}}}dv',$
${\text{Lap }}\omega$	$=\iiint {\text{lap }}\omega 'dv'$	$=\iiint {\frac {\rho '-\rho }{[\rho '-\rho ]_{0}^{3}}}\times \omega '\,dv',$

where the $\rho$ is to be regarded as constant in the integration. This extends over all space, or wherever the $u'$ or $\omega '$ have any values other than zero. These integrals may themselves be called (integral) potentials, Newtonians, and Laplacians.

This will be evident with respect both to scalar and to vector functions, if we suppose that when we differentiate the potential with respect to $x$ (thus varying the position of the point for which the potential is taken) each element of volume $dv'$ in the implied integral remains fixed, not in absolute position, but in position relative to the point for which the potential is taken. This supposition is evidently allowable whenever the integration indicated by the symbol ${\text{Pot}}$ tends to a definite limit when the limits of integration are indefinitely extended.

Since we may substitute $y$ and $z$ for $x$ in the preceding formula, and since a constant factor of any kind may be introduced under the sign of integration, we have

${\begin{aligned}\nabla {\text{ Pot }}u&={\text{ Pot }}\nabla u,\\\nabla .{\text{ Pot }}\omega &={\text{ Pot }}\nabla .\omega ,\\\nabla \times {\text{ Pot }}\omega &={\text{ Pot }}\times \omega ,\\\nabla .\nabla {\text{ Pot }}u&={\text{ Pot }}\nabla .\nabla u,\\\nabla .\nabla {\text{ Pot }}\omega &={\text{ Pot }}\nabla .\nabla \omega ,\end{aligned}}$

i.e., the symbols $\nabla ,\nabla .,\nabla \times ,\nabla .\nabla$ may be applied indifferently before or after the sign ${\text{Pot}}.$

Yet a certain restriction is to be observed. When the operation of taking the (integral) potential does not give a definite finite value, the first members of these equations are to be regarded as entirely indeterminate, but the second members may have perfectly definite values. This would be the case, for example, if $u$ or $\omega$ had a constant value throughout all space. It might seem harmless to set an indefinite expression equal to a definite, but it would be dangerous, since