As in the case of cylindrical coordinates there are many particular solutions. There are a few basic important solutions and the rest are given in terms of powers of and Legendre polynomials in .

- An isolated point source (point charge, magnetic monopole,
point source or sink) at the origin has

- If the point source is located at
then the
potential at a general point
is

- A uniform field at large in the -direction is

- A dipole of moment , orientated in the
direction has a potential

To obtain the general solutions, we look for seperable solutions along
the lines of the cylindrical case. Hence, to solve (5.27) we set

Thus, we obtain

Rearranging by dividing by gives,

The separation constant is taken as where is an integer. Thus, the radial variations satisfy

It is straightforward to show that solutions to this are given by

The variations are given by solving

This equation may be simplified by changing the independent variable from to

Thus,

Hence our equation for becomes

and the solution is given in terms of Legendre polynomials of order ,

Finally, the potential may be expressed as either

Like the trigonometric functions in the cylindrical case, the Legendre polynomials are orthogonal since,

The first two Legendre polynomials give

- so that

- and

__Example 5. .6__Find the incompressible, irrotational flow of an ideal fluid or
plasma, uniform at infinity, past a sphere of radius . The general
solution that is even in and uniform at infinity is

On the surface of the sphere we require that the radial velocity component vanishes. Thus,

This means that

From the orthogonality condition on the Legendre polynomials every coefficient must be zero, giving,

Thus,