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,

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

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

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

Then