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

98.] 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

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.

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

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 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