Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/60

and, if the space is periphractic, that the surface-integral of $u\omega$ vanishes for each of the bounding surfaces.

The existence of the minimum requires that

$\iiint u\omega .\delta \omega \,dv=0,$

while $\delta \omega$ is subject to the conditions that

$\nabla \times \delta \omega =0,$

and that the tangential component of $\delta \omega$ in the bounding surface vanishes. In virtue of these conditions we may set

$\delta \omega =\nabla \delta q,$

where $\delta q$ is an arbitrary infinitesimal scalar function of position, subject only to the condition that it is constant in each of the bounding surfaces. (See No. 67.) By substitution of this value we obtain

$\iiint u\omega .\nabla \,\delta q\,dv=0,$

or integrating by parts (No. 76)

$\iint u\omega .d\sigma \,\delta q-\iiint \nabla .[u\omega ]\delta q\,dv=0.$

Since $\delta q$ is arbitrary in the volume-integral, we have throughout the whole space

$\nabla .[u\omega ]=0;$

and since $\delta q$ has an arbitrary constant value in each of the bounding surfaces (if the boundary of the space consists of separate parts), we have for each such part

$\iint u\omega .d\sigma =0.$

Potentials, Newtonians, Laplacians.

91. Def.—If $u'$ is the scalar quantity of something situated at a certain point $\rho ',$ the potential of $u'$ for any point $\rho$ is a scalar function of $\rho ,$ defined by the equation

${\text{pot }}u'={\frac {u'}{[\rho '-\rho ]_{0}}},$

and the Newtonian of $u'$ for any point $\rho$ is a vector function of $\rho$ defined by the equation

${\text{new }}u'={\frac {\rho '-\rho }{[\rho '-\rho ]_{0}^{3}}}u'.$

Again, if $\omega '$ is the vector representing the quantity and direction of something situated at the point $\rho ',$ the potential and the Laplacian of $\omega '$ for any point $\rho$ are vector functions of $\rho$ defined by the equations

${\begin{aligned}{\text{pot }}\omega '&={\frac {\omega '}{[\rho '-\rho ]_{0}}},\\{\text{lap }}\omega '&={\frac {\rho '-\rho }{[\rho '-\rho ]_{0}^{3}}}\times \omega '.\end{aligned}}$