Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/63

and the region $S$ is a periphractic region, having within it other regions which it completely encloses.

If within any of these enclosed regions, $S_{1}$ the solenoidal condition is not fulfilled, let

$Q_{1}=\iint R\cos \epsilon \,dS_{1}$

be the surface-integral for the surface enclosing this region, and let $Q_{2},Q_{3}$ , &c. be the corresponding quantities for the other en closed regions.

Then, if a closed surface $S^{'}$ is drawn within the region $S_{1}$ the value of its surface-integral will be zero only when this surface $S^{'}$ does not include any of the enclosed regions $S_{1},S_{2}$, &c. If it includes any of these, the surface-integral is the sum of the surface integrals of the different enclosed regions which lie within it.

For the same reason, the surface-integral taken over a surface bounded by a closed curve is the same for such surfaces only bounded by the closed curve as are reconcileable with the given surface by continuous motion of the surface within the region $S$.

When we have to deal with a periphractic region, the first thing to be done is to reduce it to an aperiphractic region by drawing lines joining the different bounding surfaces. Each of these lines, provided it joins surfaces which were not already in continuous connexion, reduces the periphractic number by unity, so that the whole number of lines to be drawn to remove the periphraxy is equal to the periphractic number, or the number of internal surfaces. When these lines have been drawn we may assert that if the solenoidal condition is fulfilled in the region $S$, any closed surface drawn entirely within $S$, and not cutting any of the lines, has its surface-integral zero.

In drawing these lines we must remember that any line joining surfaces which are already connected does not diminish the periphraxy, but introduces cyclosis.

The most familiar example of a periphractic region within which the solenoidal condition is fulfilled is the region surrounding a mass attracting or repelling inversely as the square of the distance.

In this case we have

$X=m{\frac {x}{r^{3}}},\qquad X=m{\frac {y}{r^{3}}},\qquad X=m{\frac {z}{r^{3}}}$;

where $m$ is the mass supposed to be at the origin of coordinates.

At any point where $r$ is finite

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