Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/206

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$\begin{array}{ll} Q_{i} & =\mu^{i}-\frac{i(i-1)}{2.2}\mu^{i-2}\nu^{2}+\frac{i(i-1)(i-2)(i-3)}{2.2.4.4}\mu^{i-4}\nu^{4}-\mathrm{etc}.\\ \\ & \sum_{n}\left\{ (-1)^{n}\frac{|\underline{i}}{2^{2n}|\underline{n}\ |\underline{n}\ |\underline{i-2n}}\mu^{i-2n}\nu^{2n}\right\} .\end{array}$

(30)

In this expansion the coefficient of $\mu_i$ is unity, and all the other terms involve $\nu$. Hence at the pole, where $\mu=1$ and $\nu=0$, $Q_{i}=1$.

It is shewn in treatises on Laplace’s Coefficients that $Q_i$ is the coefficient of $h^i$ in the expansion of $\left(1-2\mu h+h^{2}\right)^{-\frac{1}{2}}$.

The other harmonics of the symmetrical system are most conveniently obtained by the use of the imaginary coordinates given by Thomson and Tait, Natural Philosophy, vol. i. p. 148,

$\xi=x+\sqrt{-1}y,\ \eta=x-\sqrt{-1}y.$

(31)

The operation of differentiating with respect to a axes in succession, whose directions make angles $\tfrac{\pi}{\sigma}$ with each other in the plane of the equator, may then be written

$\frac{d^{\sigma}}{dh_{1}\dots dh_{\sigma}}=\frac{d^{\sigma}}{d\xi^{\sigma}}+\frac{d^{\sigma}}{d\eta^{\sigma}}.$

(32)

The surface harmonic of degree $i$ and type $\sigma$ is found by differentiating $\tfrac{1}{r}$ with respect to $i$ axes, $\sigma$ of which are at equal intervals in the plane of the equator, while the remaining $i-\sigma$ coincide with that of $z$, multiplying the result by $r^{i+1}$ and dividing by $|\underline{i}$. Hence

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

(33)

$=(-1)^{i-s}\frac{|\underline{2s}}{2^{2s}|\underline{i}\ |\underline{s}}\left(\xi^{\sigma}+\eta^{\sigma}\right)r^{i+1}\frac{d^{i-\sigma}}{dz^{u-\sigma}}\frac{1}{r^{2\sigma+1}}.$

(34)

Now

$\xi^{\sigma}+\eta^{\sigma}=2r^{\sigma}\nu^{\sigma}\cos(\sigma\phi+\beta),$

(35)

and

$\frac{d^{i-\sigma}}{dz^{i-\sigma}}\frac{1}{r^{2\sigma+1}}=(-1)^{i-\sigma}\frac{|\underline{i+\sigma}}{2\sigma}\frac{1}{r^{i+\sigma+1}}\vartheta_{i}^{(\sigma)}.$

(36)

Hence

$Y_{i}^{(\sigma)}=2\frac{|\underline{i+\sigma}}{2^{2\sigma}|\underline{i}\ |\underline{\sigma}}\vartheta_{i}^{(\sigma)}\cos(\sigma\phi+\beta),$

(37)

where the factor 2 must be omitted when $\sigma=0$.

The quantity $\vartheta_{i}^{(\sigma)}$ is a function of $\theta$, the value of which is given in Thomson and Tait’s Natural Philosophy, vol. i. p. 149.

It may be derived from $Q_i$ by the equation

$\vartheta_{i}^{(\sigma)}=2^{\sigma}\frac{|\underline{i-\sigma}\ |\underline{\sigma}}{|\underline{i+\sigma}}\nu^{\sigma}\frac{d^{\sigma}}{d\mu^{\sigma}}Q_{i},$

(38)

where $Q_i$ is expressed as a function of $\mu$ only.