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

Rh 84. If throughout a certain space (which need not be continuous, and which may extend to infinity) and in all the bounding surfaces  and at infinite distances within the space (if such there are)  then throughout the space

86. If throughout a certain space (which need not be continuous, and which may extend to infinity) 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  and in each continuous part of which the space consists

86. If throughout any continuous space (or in all space) and in any finite part of that space, or in any finite surface in or bounding it,  then throughout the whole space  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 space contained within finite boundaries but not necessarily continuous and in all the bounding surfaces the tangential components of $$\tau$$ and $$\omega$$ are equal, then throughout the space  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,