Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/216

$\iint \left(Y_{i}^{(\sigma )}\right)^{2}dS={\frac {8\pi a^{2}}{2i+1}}\cdot {\frac {|{\underline {i+\sigma }}\ |{\underline {i-\sigma }}}{2^{2\sigma }|{\underline {i}}\ |{\underline {i}}}},$

(66)

except when $\sigma =0$, in which case we have, by equation (60),

$\iint \left(Q_{i}\right)^{2}dS={\frac {8\pi a^{2}}{2i+1}}.$

These expressions give the value of the surface-integral of the square of any surface harmonic of the symmetrical system.

We may deduce from this the value of the integral of the square of the function $\vartheta _{i}^{(\sigma )}$, given in Art. 132,

$\int _{-1}^{+1}\left(\vartheta _{i}^{(\sigma )}\right)^{2}d\mu ={\frac {2}{2i+1}}{\frac {2^{2\sigma }|{\underline {i-\sigma }}\left(|{\underline {\sigma }}\right)^{2}}{|{\underline {i+\sigma }}}}.$

(68)

This value is identical with that given by Thomson and Tait, and is true without exception for the case in which $\sigma =0$.

142.] The spherical harmonics which I have described are those of integral degrees. To enter on the consideration of harmonics of fractional, irrational, or impossible degrees is beyond my purpose, which is to give as clear an idea as I can of what these harmonics are. I have done so by referring the harmonic, not to a system of polar coordinates of latitude and longitude, or to Cartesian coordinates, but to a number of points on the sphere, which I have called the Poles of the harmonic. Whatever be the type of a harmonic of the degree $i$, it is always mathematically possible to find $i$ points on the sphere which are its poles. The actual calculation of the position of these poles would in general involve the solution of a system of $2i$ equations of the degree $i$. The conception of the general harmonic, with its poles placed in any manner on the sphere, is useful rather in fixing our ideas than in making calculations. For the latter purpose it is more convenient to consider the harmonic as the sum of $2i+1$ conjugate harmonics of selected types, and the ordinary symmetrical system, in which polar coordinates are used, is the most convenient. In this system the first type is the zonal harmonic $Q_{i}$, in which all the axes coincide with the axis of polar coordinates. The second type is that in which $i-1$ of the poles of the harmonic coincide at the pole of the sphere, and the remaining one is on the equator at the origin of longitude. In the third type the remaining pole is at 90° of longitude.

In the same way the type in which $i-\sigma$ poles coincide at the pole of the sphere, and the remaining a are placed with their axes