Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/202

We may call a point of the second degree a Quadruple point, because it is constructed by making four points approach each other. It has two axes, $h_{1}$ and $h_{2}$, and a moment $M_{2}$. The directions of these two axes and the magnitude of the moment completely define the nature of the point.

130.] Let us now consider an infinite point of degree $i$ having $i$ axes, each of which is defined by a mark on a sphere or by two angular coordinates, and having also its moment $M_{i}$, so that it is defined by $2i+1$ independent quantities. Its potential is obtained by differentiating $V_{0}$ with respect to the $i$ axes in succession, so that it may be written

$V_{i}=(-1)^{i}M_{i}{\frac {d^{i}}{dh_{1}\dots dh_{i}}}\cdot {\frac {1}{r}}.$

(13)

The result of the operation is of the form

$V_{i}=|{\underline {i}}M_{i}{\frac {Y_{i}}{r^{i+1}}},$

(14)

where $Y_{1}$ it which is called the Surface Harmonic, is a function of the $i$ cosines, $\lambda _{i}\dots \lambda _{i}$ of the angles between $r$ and the $i$ axes, and of the ${\tfrac {1}{2}}i(i-1)$ cosines, $\mu _{12}$, &c. of the angles between the different axes themselves. In what follows we shall suppose the moment $M_{i}$ unity.

Every term of $Y_{i}$ consists of products of these cosines of the form

$\mu _{12}\cdot \mu _{34}\dots \mu _{2s-1\cdot 2s}\lambda _{2s+1}\dots \lambda _{i},$

in which there are $s$ cosines of angles between two axes, and $i-2s$ cosines of angles between the axes and the radius vector. As each axis is introduced by one of the $i$ processes of differentiation, the symbol of that axis must occur once and only once among the suffixes of these cosines.

Hence in every such product of cosines all the indices occur once, and none is repeated.

The number of different products of $s$ cosines with double suffixes, and $i-2s$ cosines with single suffixes, is

$N={\frac {|{\underline {i}}}{2^{s}\ {\underline {s}}\ |{\underline {i-2s}}}}$

(15)

For if we take any one of the $N$ different terms we can form from it $2^{s}$ arrangements by altering the order of the suffixes of the cosines with double suffixes. From any one of these, again, we can form $|{\underline {s}}$ arrangements by altering the order of these cosines, and from any one of these we can form $|{\underline {i-2s}}$ arrangements by altering the order of the cosines with single suffixes. Hence, without altering the value of the term we may write it in $2^{s}\ {\underline {s}}\ |{\underline {i-2s}}$