Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/200

When $r$ is indefinitely diminished $V'$ remains finite, so that ultimately

$Vr=e.\,$

129.] There are other kinds of singular points, the properties of which we shall now investigate, but, before doing so, we must define some expressions which we shall find useful in emancipating our ideas from the thraldom of systems of coordinates.

An axis is any definite direction in space. We may suppose it defined in Cartesian coordinates by its three direction-cosines l, m, n, or, better still, we may suppose a mark made on the surface of a sphere where the radius drawn from the centre in the direction of the axis meets the surface. We may call this point the Pole of the axis. An axis has therefore one pole only, not two.

If through any point x, y, z a plane be drawn perpendicular to the axis, the perpendicular from the origin on the plane is

$p=lx+my+nz.\,$

(2)

The operation

${\frac {d}{dh}}=l{\frac {d}{dx}}+m{\frac {d}{dy}}+n{\frac {d}{dz}},$

(3)

is called Differentiation with respect to an axis $h$ whose direction-cosines are l, m, n.

Different axes are distinguished by different suffixes.

The cosine of the angle between the vector $r$ and any axis $h_{i}$ is denoted by $\lambda _{i}$ and the vector resolved in the direction of the axis $p_{i}$, where

$\lambda _{i}=l_{i}x+m_{i}y+n_{i}z=p_{i}.$

(4)

The cosine of the angle between two axes $h_{i}$ and $h_{j}$ is denoted by $\mu _{ij}$ where

$\mu _{ij}=l_{i}l_{j}+m_{i}m_{j}+n_{i}n_{j}.$

(5)

From these definitions it is evident that

${\frac {dr}{dh_{i}}}={\frac {p_{i}}{r}}=\lambda _{i},$

(6)

${\frac {dp_{j}}{dh_{i}}}=\mu _{ij}={\frac {dp_{i}}{dh_{j}}},$

(7)

${\frac {d\lambda _{i}}{dh_{j}}}={\frac {\mu _{ij}-\lambda _{i}\lambda _{j}}{r}}$

(8)

Now let us suppose that the potential at the point (x, y, z) due to a singular point of any degree placed at the origin is

$Mf(x,y,z).\,$

If such a point be placed at the extremity of the axis $h$, the potential at (x, y, z) will be

$Mf\left((x-lh),\ (y-mh),\ (z-nh)\right)\,;$