Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/62

22 and we know that

therefore

or, in other words, the surface-integral over the surface $$S_2$$ is equal and opposite to that over $$S_1$$ whatever be the form and position of $$S_2$$, provided that the intermediate surface $$S_0$$ is one for which $$R$$ is always tangential.

If we suppose $$L_1$$ a closed curve of small area, $$S_0$$ will be a tubular surface having the property that the surface-integral over every complete section of the tube is the same.

Since the whole space can be divided into tubes of this kind provided

a distribution of a vector quantity consistent with this equation is called a Solenoidal Distribution.

On Tubes and Lines of Flow.

If the space is so divided into tubes that the surface-integral for every tube is unity, the tubes are called Unit tubes, and the surface-integral over any finite surface $$S$$ bounded by a closed curve $$L$$ is equal to the number of such tubes which pass through $$S$$ in the positive direction, or, what is the same thing, the number which pass through the closed curve $$L$$.

Hence the surface-integral of $$S$$ depends only on the form of its boundary $$L$$, and not on the form of the surface within its boundary.

On Periphractic Regions.

If, throughout the whole region bounded externally by the single closed surface $$S_1$$ the solenoidal condition

$$\frac {dX} {dx}+ \frac {dY} {dy} + \frac {dZ} {dz} = 0$$

is fulfilled, then the surface-integral taken over any closed surface drawn within this region will be zero, and the surface-integral taken over a bounded surface within the region will depend only on the form of the closed curve which forms its boundary.

It is not, however, generally true that the same results follow if the region within which the solenoidal condition is fulfilled is bounded otherwise than by a single surface.

For if it is bounded by more than one continuous surface, one of these is the external surface and the others are internal surfaces,