Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/155

Rh found for a particular position of $$P$$. To find the form of the function when the form of the surface is given and the position of $$P$$ is arbitrary, is a problem of far greater difficulty, though, as we have proved, it is mathematically possible.

Let us suppose the problem solved, and that the point $$P$$ is taken within the surface. Then for all external points the potential of the superficial distribution is equal and opposite to that of $$P$$. The superficial distribution is therefore centrobaric, and its action on all external points is the same as that of a unit of negative electricity placed at $$P$$.

102.] Let a region be completely bounded by a number of surfaces $$S_{0},S_{1},S_{2}$$, &c., and let $$K$$ be a quantity, positive or zero but not negative, given at every point of this region. Let $$V$$ be a function subject to the conditions that its values at the surfaces $$S_{1},S_{2}$$, &c. are the constant quantities $$C_{1},C_{2}$$, &c., and that at the surface $$S_0$$

where $$\nu$$ is a normal to the surface $$S_0$$. Then the integral

taken over the whole region, has a unique minimum when $$V$$ satisfies the equation

throughout the region, as well as the original conditions.

We have already shewn that a function $$V$$ exists which fulfils the conditions (1) and (3), and that it is determinate in value. We have next to shew that of all functions fulfilling the surface-conditions it makes $$Q$$ a minimum.

Let $$V_0$$ be the function which satisfies (1) and (3), and let

be a function which satisfies (1).

It follows from this that at the surfaces $$S_{1},S_{2}$$, &c. $$U=0$$.

The value of $$Q$$ becomes