Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/205

have made the conception of the general spherical harmonic of any integral degree perfectly definite to myself, and I hope also to those who may have felt the vagueness of some other forms of the expression.

When the poles are given, the value of the harmonic for a given point on the sphere is a perfectly definite numerical quantity. When the form of the function, however, is given, it is by no means so easy to find the poles except for harmonics of the first and second degrees and for particular cases of the higher degrees.

Hence, for many purposes it is desirable to express the harmonic as the sum of a number of other harmonics, each of which has its axes disposed in a symmetrical manner.

Symmetrical System.

132.] The particular forms of harmonics to which it is usual to refer all others are deduced from the general harmonic by placing $i-\sigma$ of the poles at one point, which we shall call the Positive Pole of the sphere, and the remaining $\sigma$ poles at equal distances round one half of the equator.

In this case $\lambda _{1},\ \lambda _{2},\ \dots \lambda _{i-\sigma }$ are each of them equal to $\cos \theta$, and $\lambda _{i-s+1}\dots \lambda _{i}$ are of the form $\sin \theta \cos(\phi -\beta )$. We shall write $\mu$ for $\cos \theta$ and $\nu$ for $\sin \theta$.

Also the value of $\mu _{jj'}$ is unity if $j$ and $j'$ are both less than $i-\sigma$, zero when one is greater and the other less than this quantity, and $\cos n{\frac {\pi }{\sigma }}$ when both are greater.

When all the poles are concentrated at the pole of the sphere, the harmonic becomes a zonal harmonic for which $\sigma =0$. As the zonal harmonic is of great importance we shall reserve for it the symbol $Q_{i}$.

We may obtain its value either from the trigonometrical expression (27), or more directly by differentiation, thus

$Q_{i}=(-1)^{i}{\frac {r^{i+1}}{|{\underline {i}}}}{\frac {d^{i}}{dz^{i}}}\left({\frac {1}{r}}\right),$

(28)

${\begin{array}{c}Q_{i}={\frac {1.3.5\dots (2i-1)}{1.2.3\dots i}}\left\{\mu ^{i}-{\frac {i(i-1)}{2.(2i-1)}}\mu ^{i-2}+{\frac {i(i-1)(i-2)(i-3)}{2.4.(2i-1)(2i-3)}}\mu ^{i-4}-\mathrm {etc} .\right\}\\\\=\sum _{n}\left\{(-1)^{n}{\frac {|{\underline {2i-2n}}}{2^{i}|{\underline {n}}\ |{\underline {i-n}}|{\underline {i-2n}}}}\mu ^{i-2n}\right\}\end{array}}$

(29)

It is often convenient to express $Q_{i}$ as a homogeneous function of $\cos \theta$ and $\sin \theta$, which we shall write $\mu$, and $\nu$ respectively,