A sphere of radius \(a\) has centre \(O\), and \(P\) is a point distant \(z\) from \(O\). Find the mean value with respect to area of the \(n\)th power of the distance of the surface of the sphere from \(P\), where \(n \ge -1\) and is not necessarily integral, distinguishing between the cases when \(z > a\) and \(z < a\). Verify that if \(P\) is external to the sphere and \(P'\) is the inverse of \(P\) with respect to the sphere, the mean value for \(P'\) is \((\frac{a}{z})^n\) that for \(P\).