Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/177

or the line in which they meet is a point or line of equilibrium of the $n$th degree.

When $C$ becomes equal to $V_{1}$ the positive region is reduced to the electrified point or conductor of highest potential, and has therefore lost all its periphraxy. Hence, if each point or line of equilibrium counts for one, two, or $n$ according to its degree, the number so made up by the points or lines now considered will be one less than the number of negatively electrified bodies.

There are other points or lines of equilibrium which occur where the positive regions become separated from each other, and the negative region acquires periphraxy. The number of these, reckoned according to their degrees, is one less than the number of positively electrified bodies.

If we call a point or line of equilibrium positive when it is the meeting-place of two or more positive regions, and negative when the regions which unite there are negative, then, if there are $p$ bodies positively and $n$ bodies negatively electrified, the sum of the degrees of the positive points and lines of equilibrium will be $p-1$, and that of the negative ones $n-1$.

But, besides this definite number of points and lines of equilibrium arising from the junction of different regions, there may be others, of which we can only affirm that their number must be even. For if, as the negative region expands, it meets itself, it becomes a cyclic region, and it may acquire, by repeatedly meeting itself, any number of degrees of cyclosis, each of which corresponds to the point or line of equilibrium at which the cyclosis was established. As the negative region continues to expand till it fills all space, it loses every degree of cyclosis it has acquired, and becomes at last acyclic. Hence there is a set of points or lines of equilibrium at which cyclosis is lost, and these are equal in number of degrees to those at which it is acquired. .

If the form of the electrified bodies or conductors is arbitrary, we can only assert that the number of these additional points or lines is even, but if they are electrified points or spherical conductors, the number arising in this way cannot exceed ($n-l$)($n-2$), where $n$ is the number of bodies.

114.] The potential close to any point $P$ may be expanded in the series

$V=V_{0}+H_{1}+H_{2}+\mathrm {etc} .;$

where $H_{1},\ H_{2}$, &c. are homogeneous functions of $x,y,z$, whose dimensions are 1, 2, &c. respectively.