Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/334

302 where the orders of all the harmonics are odd.

696.] Let us begin by supposing the two magnetic shells which are equivalent to the currents to be portions of two concentric spheres, their radii being $$c_1$$ and $$c_2$$, of which $$c_1$$ is the greater (Fig. 47). Let us also suppose that the axes of the two shells coincide, and that $$a_1$$ the angle subtended by the radius of the first shell, and $$a_2$$ the angle subtended by the radius of the second shell at the centre $$C$$.

Let $$\omega_1$$ be the potential due to the first shell at any point within it, then the work required to carry the second shell to an infinite distance is the value of the surface-integral Rhextended over the second shell. Hence Rh Rh

or, substituting the value of the integrals from equation (2), Art. 694,{{rh||$$M = 4\pi^2 \sin^2 \alpha_1 \sin^2 \alpha_2 c_2^2 \left \{ \frac{1}{2} \frac{c_2}{c_1} Q_1^\prime(\alpha_1) Q_1^\prime(\alpha_2) + \mathrm{\&c.} + \frac{1}{i(i+1)} \frac{c_2^i}{c_1^i} Q_i^\prime (\alpha_1) Q_i^\prime )a_2)\right.$$|}}