Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/214

${\begin{array}{ll}\iint Y_{i}Q_{j}dS&={\frac {4\pi a^{2}}{(2i+1)|{\underline {i}}}}S\left\{(-1)^{s}{\frac {|{\underline {2i-2s}}}{2^{i-s}|{\underline {i-s}}}}\sum \left(\lambda ^{i-2s}\mu ^{s}\right)\right\},\\\\&={\frac {4\pi a^{2}}{2i+1}}Y_{i(j)},\end{array}}$

(60)

where $Y_{i(j)}$ denotes the value of $Y_{i}$ in equation (27) at the common pole of all the axes of $Q_{j}$.

140.] This result is a very important one in the theory of spherical harmonics, as it leads to the determination of the form of a series of spherical harmonics, which expresses a function having any arbitrarily assigned value at each point of a spherical surface.

For let $F$ be the value of the function at any given point of the sphere, say at the centre of gravity of the element of surface $dS$, and let $Q_{i}$ be the zonal harmonic of degree $i$ whose pole is the point $P$ on the sphere, then the surface-integral

$\iint FQ_{i}dS$

extended over the spherical surface will be a spherical harmonic of degree $i$, because it is the sum of a number of zonal harmonics whose poles are the various elements $dS$, each being multiplied by $FdS$. Hence, if we make

$A_{i}Y_{i}={\frac {2i+1}{4\pi a^{2}}}\iint FQ_{i}dS,$

(61)

we may expand F in the form

$F=A_{0}Y_{0}+A_{1}Y_{1}+\mathrm {etc} .+A_{i}Y_{i},$

(62)

or

$F={\frac {1}{4\pi a^{2}}}\left\{\iint FQ_{0}dS+3\iint FQ_{1}dS+\mathrm {etc} .+(2i+1)\iint fQ_{i}dS\right\}.$

(63)

This is the celebrated formula of Laplace for the expansion in a series of spherical harmonics of any quantity distributed over the surface of a sphere. In making use of it we are supposed to take a certain point $P$ on the sphere as the pole of the zonal harmonic $Q_{i}$, and to find the surface-integral

$\iint FQ_{i}dS$

over the whole surface of the sphere. The result of this operation when multiplied by $2i+1$ gives the value of $A_{i}Y_{i}$ at the point $P$, and by making $P$ travel over the surface of the sphere the value of $A_{i}Y_{i}$ at any other point may be found.