The determination of the magnetic moment of a magnet.

It involves the determination by experiment of–( a) the product of the
magnetic moment, M, of the magnet by the horizontal component, H, of the
earth’s magnetism; (b) the quotient of M divided by H. Knowing these two
quantities, M is given by the formula M = SquareRoot( )M * H) * (M/H) )
and if desired H is given by the formula H = SquareRoot( (M*H) / (M/H)).

M*H is determined by the method of vibrations. A very long, thin magnet
suspended by a torsion filament is caused to oscillate, and its period
is determined. Calling such period T and the moment of inertia of the
magnet I, we have the formula T= 2* PI * SquareRoot( I / (H*M) )  (a),
whence H*M is calculated, I of course being known or separately
determined.

M/H is determined by the End-on deflection method, or the Broadside
deflection method. In both cases the deflection of a compass needle by
the magnet in question is the basis of the work.

In the end-on method AB is the magnet under examination; DE the compass
needle; a the angle of deflection; d  the distance between C and the
middle of AB, which should be considerable compared with the length of
DE; 2l, the length of AB. We then have the formula
tan a = (M/H) * (2d / (d^2 – l^2)^2),
which if 2l is small compared to d reduces to
tan a = M/Hd 3

(b), which gives M/H, a and d being known.

In the broadside method the line d is the magnetic meridian, and the
diagram shows the relative positions. We then have the formula
tan a = (M/H) / (d2 + l2)^1.5;
which if 1 is relatively small reduces to
tan a = M/(H * d3 )(C.)

a and c or a and b can be combined giving M and H in C.G.S. measurement.