Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/212

The number of different products for a given value of $s$ is

$N={\frac {\left(|{\underline {i}}\right)^{2}}{2^{2s}\left(|{\underline {s}}\right)^{2}|{\underline {i-2s}}}}.$

(55)

The final result is easily obtained by the successive differentiation of

$r_{j}Y_{j}={\frac {1}{|{\underline {j}}}}S\left\{(-1)^{s}{\frac {|{\underline {2j-2s}}}{2^{j-s}|{\underline {j-s}}}}r^{2s}\sum \left(p^{j-2s}\mu ^{s}\right)\right\}.$

Differentiating this $i$ times in succession with respect to the new axes, so as to obtain any given combination of the axes in pairs, we find that in differentiating $r^{2s}$ with respect to $s$ of the new axes, which are to be combined with other axes of the new system, we introduce the numerical factor $2s(2s-2)\dots 2$, or $2^{s}|{\underline {s}}$. In continuing the differentiation the $p$’s become converted into $\mu$’s, but no numerical factor is introduced. Hence

${\frac {d^{i}}{dh_{1}\dots dh_{i}}}r^{j}Y_{j}={\frac {1}{|{\underline {i}}}}S\left\{(-1)^{s}{\frac {|{\underline {2i-2s}}\ |{\underline {s}}}{2^{i-s}|{\underline {i-s}}}}\sum \left(\mu _{ii}^{s}\mu _{jj}^{s}\mu _{ij}^{i-2s}\right)\right\}$

(56)

Substituting this result in equation (54) we find for the value of the surface-integral of the product of two surface harmonics of the same degree, taken over the surface of a sphere of radius $a$,

$\iint Y_{i}Y_{j}dS={\frac {4\pi a^{2}}{(2i+1)\left(|{\underline {i}}\right)^{2}}}S\left\{(-1)^{s}{\frac {|{\underline {2i-2s}}\ |{\underline {s}}}{2^{i-s}|{\underline {i-s}}}}\sum \left(\mu _{ii}^{s}\mu _{jj}^{s}\mu _{ij}^{i-2s}\right)\right\}$

(57)

This quantity differs from zero only when the two harmonics are of the same degree, and even in this case, when the distribution of the axes of the one system bears a certain relation to the distribution of the axes of the other, this integral vanishes. In this case, the two harmonics are said to be conjugate to each other.

On Conjugate Harmonics.

138.] If one harmonic is given, the condition that a second harmonic of the same degree may be conjugate to it is expressed by equating the right hand side of equation (57) to zero.

If a third harmonic is to be found conjugate to both of these there will be two equations which must be satisfied by its $2i$ variables.

If we go on constructing new harmonics, each of which is conjugate to all the former harmonics, the variables will be continually more and more restricted, till at last the $(2i+1)$th harmonic will have all its variables determined by the $2i$ equations, which must