MM′ (sin θ cos φ − 2 sin φ cos θ) / r³.
(18)
This vanishes if sin θ cos φ = 2 sin φ cos θ, i.e. if tan φ = ½ tan θ, S′N′ being then along a line of force, a result which explains the construction given above for finding the direction of the force F in (14). If the axis of SN produced passes through the centre of S′N′, θ = 0, and the couple becomes
2MM′ sin φ/r³,
(19)
tending to diminish φ; this is called the “end on” position. If the centre of S′N′ is on the perpendicular bisector of SN, θ = ½π, and the couple will be
MM′ cos φ/r³,
(20)
tending to increase φ; this is the “broadside on” position. These two positions are sometimes called the first and second (or A and B) principal positions of Gauss. The components X, Y, parallel and perpendicular to r, of the force between the two magnets SN and S′N′ are
X = 3MM′ (sin θ sin φ − 2 cos θ cos φ) / r4,