Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/333

Rh Rh

This expression fails when $$i = 0$$, but since $$Q_0 = 1$$, Rh

As the function $$\frac{dQ_i}{d\mu}$$ occurs in every part of this investigation we shall denote it by the abbreviated symbol $$Q_i^\prime$$. The values of $$Q_i^\prime$$ corresponding to several values of $$i$$ are given in Art. 698.

We are now able to write down the value of $$P$$ for any point $$R$$, whether on the axis or not, by substituting $$r$$ for $$z$$, and multiplying each term by the zonal harmonic of $$\theta$$ of the same order. For $$P$$ must be capable of expansion in a series of zonal harmonics of $$\theta$$ with proper coefficients. When $$\theta = 0$$ each of the zonal harmonics becomes equal to unity, and the point $$R$$ lies on the axis. Hence the coefficients are the terms of the expansion of $$P$$ for a point on the axis. We thus obtain the two series

695.] We may now find $$\omega$$, the magnetic potential of the circuit, by the method of Art. 670, from the equation Rh

We thus obtain the two series

The series (6) is convergent for all values of $$r$$ less than $$c$$, and the series (6′) is convergent for all values of $$r$$ greater than $$c$$. At the surface of the sphere, where $$r = c$$, the two series give the same value for $$\omega$$ when $$\theta$$ is greater than $$\alpha$$, that is, for points not occupied by the magnetic shell, but when $$\theta$$ is less than $$\alpha$$, that is, at points on the magnetic shell, Rh

If we assume $$O$$, the centre of the circle, as the origin of coordinates, we must put $$\alpha =\frac{\pi}{2}$$, and the series become