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

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where is a function of the direction of , 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 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 from equation (74) in these series.

Hence, if is the value of the potential of the conductor and the charge upon it,


Since 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 , in terms of spherical harmonics. If can be expanded in terms of spherical harmonics of degrees lower than , then can be expanded in spherical harmonics of degrees lower than .

Let therefore


then the coefficients will each of them be small compared with the coefficients on account of the smallness of , and therefore the last term of equation (76), consisting of terms in , may be neglected.

Hence the coefficients of the form may be found by expanding equation (76) in spherical harmonics.

For example, let the body have a charge , and be acted on by no external force.

Let be expanded in a series of the form