Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/215

But $A_{i}Y_{i}$ is a general surface harmonic of degree $i$, and we wish to break it up into the sum of a series of multiples of the $2i+1$ conjugate harmonics of that degree.

Let $P_{i}$ be one of these conjugate harmonics of a particular type, and let $B_{i}P_{i}$ be the part of $A_{i}Y_{i}$ belonging to this type.

We must first find

$M=\iint P_{i}P_{i}dS,$

(64)

which may be done by means of equation (57), making the second set of poles the same, each to each, as the first set.

We may then find the coefficient $B_{i}$ from the equation

$B_{i}={\frac {1}{M}}\iint FP_{i}dS.$

For suppose $F$ expanded in terms of spherical harmonics, and let $B_{j}P_{j}$ be any term of this expansion. Then, if the degree of $P_{j}$ is different from that of $P_{i}$, or if, the degree being the same, $P_{j}$ is conjugate to $P_{i}$, the result of the surface-integration is zero. Hence the result of the surface-integration is to select the coefficient of the harmonic of the same type as $P_{i}$.

The most remarkable example of the actual development of a function in a series of spherical harmonics is the calculation by Gauss of the harmonics of the first four degrees in the expansion of the magnetic potential of the earth, as deduced from observations in various parts of the world.

He has determined the twenty-four coefficients of the three conjugate harmonics of the first degree, the five of the second, seven of the third, and nine of the fourth, all of the symmetrical system. The method of calculation is given in his General Theory of Terrestrial Magnetism.

141.] When the harmonic $P_{i}$ belongs to the symmetrical system we may determine the surface-integral of its square extended over the sphere by the following method.

The value of $r^{i}Y_{i}^{\sigma }$ is, by equations (34) and (36),

$r^{i}Y_{i}^{(\sigma )}={\frac {|{\underline {i}}+\sigma }{2^{\sigma }|{\underline {i}}\ |{\underline {\sigma }}}}\left(\xi ^{\sigma }+\eta ^{\sigma }\right)\left(z^{i-\sigma }-{\frac {(i-\sigma )(i-\sigma -1)}{4(\sigma +1)}}z^{i-\sigma -2}\xi \eta +\mathrm {etc} .\right);$

and by equations (33) and (54),

$\iint \left(Y_{i}^{(\sigma )}\right)^{2}dS={\frac {4\pi }{|{\underline {i}}}}{\frac {a^{2}}{2i+1}}\cdot {\frac {d^{i-\sigma }}{dz^{i-\sigma }}}\left({\frac {d^{\sigma }}{d\xi ^{\sigma }}}+{\frac {d^{\sigma }}{d\eta ^{\sigma }}}\right)\left(r_{i}Y_{i}^{(\sigma )}\right)$

Performing the differentiations, we find that the only terms which do not disappear are those which contain $z^{i-\sigma }$. Hence