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



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,

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

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

Now

and

Hence

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

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