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

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^[1], and its action on all external points is the same as that of a unit of negative electricity placed at $P$.

Method of Approximating to the Values of Coefficients of Capacity, &c.

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 $S_{0}$

${\frac {dV}{d\nu }}=0,$

(1)

where $\nu$ is a normal to the surface $S_{0}$. Then the integral

$Q={\frac {1}{8\pi }}\iiint K\left(\left({\frac {dV}{dx}}\right)^{2}+\left({\frac {dV}{dy}}\right)^{2}+\left({\frac {dV}{dz}}\right)^{2}\right)dx\ dy\ dz,$

(2)

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

${\frac {d}{dx}}K{\frac {dV}{dx}}+{\frac {d}{dy}}K{\frac {dV}{dy}}+{\frac {d}{dz}}K{\frac {dV}{dz}}=0$

(3)

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

$V=V_{0}+U\,$

(4)

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

$Q={\frac {1}{8\pi }}\iiint \left\{K\left(\left({\frac {dV_{0}}{dx}}\right)^{2}+\mathrm {\&c.} \right)+K\left(\left({\frac {dU}{dx}}\right)^{2}+\mathrm {\&c.} \right)+2K\left({\frac {dV_{0}}{dx}}{\frac {dU}{dx}}+\mathrm {\&c.} \right)\right\}dx\ dy\ dz.$

(5)

1. Thomson and Tait’s Natural Philosophy, § 526.