Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/143

Hence $M=0$ for a space bounded by a single equipotential surface.

If the space is bounded externally by the surface V = C, and internally by the surfaces $V=C_{1}$, $V=C_{2}$, &c., then the total value of $M$ for the space so bounded will be

$M-M_{1}-M_{2}\,\!$ &c.,

where $M$ is the value of the integral for the whole space within the surface $V=C$, and $M_{l},M_{2},$ &c. are the values of the integral for the spaces within the internal surfaces. But we have seen that $M,M_{1},M_{2}$, &c. are each of them zero, so that the integral is zero also for the periphractic region between the surfaces.

Case 2. If $lu+mv+nw$ is zero over any part of the bounding surface, that part of the surface can contribute nothing to the value of $M$, because the quantity under the integral sign is everywhere zero. Hence $M$ will remain zero if a surface fulfilling this con dition is substituted for any part of the bounding surface, provided that the remainder of the surface is all at the same potential.

98.] We are now prepared to prove a theorem which we owe to Sir William Thomson ^[1].

As we shall require this theorem in various parts of our subject, I shall put it in a form capable of the necessary modifications.

Let $a,b,c$ be any functions of $x,y,z$ (we may call them the components of a flux) subject only to the condition

${\frac {da}{dx}}+{\frac {db}{dy}}+{\frac {dc}{dz}}+4\pi \rho =0$

(5)

where $\rho$ has given values within a certain space. This is the general characteristic of $a,b,c$.

Let us also suppose that at certain surfaces (S) $a,b$, and $c$ are discontinuous, but satisfy the condition

$l(a_{1}-a_{2})+m(b_{1}-b_{2})+n(c_{1}-c_{2})+4\pi \rho =0\,\!$;

(6)

where $l,m,n$ are the direction-cosines of the normal to the surface, $a_{1},b_{1},c_{1}$ the values of $a,b,c$ on the positive side of the surface, and $a_{2},b_{2},c_{2}$ those on the negative side, and $\sigma$ a quantity given for every point of the surface. This condition is the superficial characteristic of $a,b,c$.

Next, let us suppose that $V$ is a continuous function of $x,y,z$, which either vanishes at infinity or whose value at a certain point is given, and let $V$ satisfy the general characteristic equation

1. Cambridge and Dublin Mathematical Journal, February, 1848.