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Suppose we have a spherical shell with charge density on the surface [itex]\sigma[/itex] and radius R. The potential inside the shell is given by:

V_(x,y,z) = [itex]\frac{V0}{R^{2}}(6z^2+ax^2+by^2)[/itex]

It is assumed, that the potential is rotational symmetric around the z-axis inside and outside the shell, and goes to 0 far away from the shell. There's no charge inside and outside the shell and no outer field.

How do i determine the constants a and b?

Mabye change to spherical coordinates and solve the equation:

[itex]\frac{\partial{V}}{\partial{\theta}}=0[/itex]

for a or b. But i can't figure out any other conditions if this is right.