Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/221

where $F$ is a function of the direction of $r$, and is a numerical quantity the square of which may be neglected.

Let the potential due to the external electrified system be expressed, as before, in a series of solid harmonics of positive degree, and let the potential $U$ be a series of solid harmonics of negative degree. Then the potential at the surface of the conductor is obtained by substituting the value of $r$ from equation (74) in these series.

Hence, if $C$ is the value of the potential of the conductor and $B_{0}$ the charge upon it,

${\begin{array}{ll}C=&A_{0}+A_{1}aY_{1}+\dots +A_{i}a^{i}Y_{i},\\\\&\qquad +A_{1}aFY_{1}+\dots +iA_{i}a^{i}FY_{i},\\\\&+B_{0}{\frac {1}{a}}+B_{1}{\frac {1}{a^{2}}}Y_{1}+\dots +B_{i}a^{-(i+1)}Y_{i}+\dots +B_{j}a^{-j+1}Y_{j},\\\\&-B_{0}{\frac {1}{a}}-2B_{1}{\frac {1}{a^{2}}}FY_{1}+\dots -(i+1)B_{i}a^{-(i+1)}FY_{i}+\dots -(j+1)B_{j}a^{-(j+1)}FY_{j}.\end{array}}$

(75)

Since $F$ is very small compared with unity, we have first a set of equations of the form (72), with the additional equation

${\begin{array}{ll}0=&-B_{0}{\frac {1}{a}}F+3A_{1}aFY_{1}+\mathrm {etc} .+(i+1)A_{i}a^{i}FY_{i}\\\\&+\sum \left(B_{j}a^{-(j+1)}Y_{j}\right)-\sum \left((j+1)B_{j}a^{-(j+1)}FY_{j}\right).\end{array}}$

(76)

To solve this equation we must expand $F$, $FY_{1}\dots FY_{i}$ in terms of spherical harmonics. If $F$ can be expanded in terms of spherical harmonics of degrees lower than $k$, then $FY_{i}$ can be expanded in spherical harmonics of degrees lower than $i+k$.

Let therefore

$B_{0}{\frac {1}{a}}F-3A_{1}aFY_{1}-\dots -(2i+1)A_{i}a^{i}FY_{i}=\sum \left(B_{j}a^{-(j+1)}Y_{j}\right),$

(77)

then the coefficients $B_{j}$ will each of them be small compared with the coefficients $B_{0}\dots B_{i}$ on account of the smallness of $F$, and therefore the last term of equation (76), consisting of terms in $B_{j}F$, may be neglected.

Hence the coefficients of the form $B_{j}$ may be found by expanding equation (76) in spherical harmonics.

For example, let the body have a charge $B_{0}$, and be acted on by no external force.

Let $F$ be expanded in a series of the form

$F=S_{1}Y_{1}+\mathrm {etc} .+S_{k}Y_{k}.$

(78)

Then

$B_{0}{\frac {1}{a}}S_{1}Y_{1}+\mathrm {etc} .+B_{0}{\frac {1}{a}}S_{k}Y_{k}=\sum \left(B_{j}a^{-(j+1)}Y_{j}\right),$

(79)