Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/142

102 If $$l, m, n$$ are the direction-cosines of the normal drawn inwards from the bounding surface at any point, and $$dS$$ an element of that  surface, then we may write

the integration of the first term being extended over the bounding surface, and that of the second throughout the entire space.

For all spaces within which $$u, v, w$$ are continuous, the second term vanishes in virtue of equation (1). If for any surface within the space $$u, v, w$$ are discontinuous but subject to equation (2), we  find for the part of $$M$$ depending on this surface,

where the suffixes $$_1$$ and $$_2$$, applied to any symbol, indicate to which of the two spaces separated by the surface the symbol belongs.

Now, since $$V$$ is continuous, we have at every point of the surface, but since the normals are drawn in opposite directions, we have

so that the total value of M, so far as it depends on the surface of discontinuity, is

The quantity under the integral sign vanishes at every point in virtue of the superficial solenoidal condition or characteristic (2).

Hence, in determining the value of $$M$$, we have only to consider the surface-integral over the actual bounding surface of the space  considered, or

Case 1. If V is constant over the whole surface and equal to $$C$$,

The part of this expression under the sign of double integration represents the surface-integral of the flux whose components are  $$u, v, w,$$ and by Art. 21 this surface-integral is zero for the closed surface in virtue of the general and superficial solenoidal conditions  (1) and (2).