Inverse $n^{th}$ power law for Magnetism experiment

Question

I'm trying to read through a lab manual for showing the inverse square law of magnetism by Gauss' Method.

The following diagram is given

where M is the magnetic moment of the test magnet. $H_0$ is the horizontal component of earth's magnetic field.

It introduces this set of equations for the "Inverse $n^{th}$ power law":

Magnetic field at position A, $$ H_A = \frac{nM}{d^{n+1}}\frac{\mu_0}{4\pi}$$

Magnetic field at position B, $$ H_B = \frac{M}{d^{n+1}}\frac{\mu_0}{4\pi}$$

From where do these set of equations come from? Alternatively: Is there a clear resource on Gauss' method for this experiment?

Answer 1

The device which is drawn in the diagram is called a deflection magnetometer which is capable of measuring the deflection of a compass needle accurately.
It was an instrument often used in student laboratories to investigate magnetism and here is an example of its use.