Schwere, Elektricität und Magnetismus:399

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

${\displaystyle \nabla t=\nabla u,\,}$

and in each continuous part of which the space consists

${\displaystyle t-\,}$[1]${\displaystyle u={\text{constant}}.\,}$

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

${\displaystyle \nabla \times \tau =\nabla \times \omega \,}$ and ${\displaystyle \nabla \cdot \tau =\nabla \cdot \omega ,\,}$

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

${\displaystyle \tau =\omega ,\,}$

then throughout the whole space

${\displaystyle \tau =\omega .\,}$

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

     87. If throughout an aperiphractic*^[2] space contained within finite boundaries but not necessarily continuous

${\displaystyle \nabla \times \tau =\nabla \times \omega \,}$ and ${\displaystyle \nabla \cdot \tau =\nabla \cdot \omega ,\,}$

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

${\displaystyle \tau =\omega .\,}$

     It is evidently sufficient to prove this proposition for a continuous space. Setting ${\displaystyle \nabla u=\tau -\omega \,}$, we have ${\displaystyle \nabla \cdot \nabla u=0\,}$ for the whole space, and ${\displaystyle u={\text{constant}}\,}$ for its boundary, which will be a single surface for a continuous aperiphractic space. Hence throughout the space ${\displaystyle \nabla u=\tau -\omega =0\,}$.

     88. If throughout an acyclic space contained within finite boundaries but not necessarily continuous

${\displaystyle \nabla \times \tau =\nabla \times \omega \,}$ and ${\displaystyle \nabla \cdot \tau =\nabla \cdot \omega ,\,}$

and in all the bounding surfaces the normal components of ${\displaystyle \tau \,}$ and ${\displaystyle \omega \,}$ are equal, then throughout the whole space

${\displaystyle \tau =\omega .\,}$