Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/57

84. If throughout a certain space (which need not be continuous, and which may extend to infinity)

$\nabla .\nabla t=\nabla .\nabla u,$

and in all the bounding surfaces

$t=u,$

and at infinite distances within the space (if such there are)

$t=u,$

then throughout the space

$t=u.$

86. If throughout a certain space (which need not be continuous, and which may extend to infinity)

$\nabla .\nabla t=\nabla .\nabla u,$

and in all the bounding surfaces the normal components of $\nabla t$ and $\nabla u$ are equal, and at infinite distances within the space (if such there are) $r^{2}\left({\frac {dt}{dr}}-{\frac {du}{dr}}\right)=0,$ where $r$ denotes the distance from some fixed origin,—then throughout the space

$\nabla t=\nabla u,$

and in each continuous part of which the space consists

$t-u={\text{constant.}}$

86. If throughout any continuous space (or in all space)

$\nabla \times \tau =\nabla \times \omega$and$\nabla .\tau =\nabla .\omega ,$

and in any finite part of that space, or in any finite surface in or bounding it,

$\tau =\omega ,$

then throughout the whole space

$\tau =\omega .$

For, since $\nabla \times (\tau -\omega )=0,$ we may set $\nabla u=\tau -\omega ,$ making the space acyclic (if necessary) by diaphragms. Then in the whole space $u$ is single-valued and $\nabla .\nabla u=0,$ and in a part of the space, or in a surface in or bounding it, $\nabla u=0.$ Hence throughout the space $\nabla u=\tau -\omega =0.$

87. If throughout an aperiphractic^[1] space contained within finite boundaries but not necessarily continuous

$\nabla \times \tau =\nabla \times \omega$and$\nabla .\tau =\nabla .\omega ,$

and in all the bounding surfaces the tangential components of $\tau$ and $\omega$ are equal, then throughout the space

$\tau =\omega .$

It is evidently sufficient to prove this proposition for a continuous space. Setting $\nabla u=\tau -\omega ,$ we have $\nabla .\nabla u=0$ for the whole space,

1. If a space encloses within itself another space, it is called periphractict, otherwise aperiphractic.