# 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)