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

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

(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.