Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/54

38 76. From equation (4) we obtain where, as elsewhere in these equations, the surface-integral relates to the boundary of the volume-integrals.

From this, by substitution of $$\nabla t$$ for $$\omega,$$ we derive as a particular case which is Green's Theorem. The substitution of $$s\nabla t$$ for $$\omega$$ gives the more general form of this theorem which is due to Thomson, viz., 77. From equation (6) we obtain A particular case is

78. If throughout any continuous space (or in all space) then throughout the same space  79. 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  This will appear from the following considerations: If $$\nabla u = 0$$ in any finite part of the space, $$u$$ is constant in that part.

If $$u$$ is not constant throughout, let us imagine a sphere situated principally in the part in which $$u$$ is constant, but projecting slightly into a part in which $$u$$ has a greater value, or else into a part in which $$u$$ has a less. The surface-integral of $$\nabla u$$ for the part of the spherical surface in the region where $$u$$ is constant will have the value zero: for the other part of the surface, the integral will be either greater than zero, or less than zero. Therefore the whole surface-integral for the spherical surface will not have the value zero, which is required by the general condition, $$\nabla. \nabla u = 0.$$

Again, if $$\nabla u = 0$$ only in a surface in or bounding the space in