For a dipolar geomagnetic field, the vertical (V) and horizontal (H) components at magnetic latitude λ relate to the equatorial total field F0 via V=2F0 sinλ and H=F0 cosλ, which combine algebraically to give the invariant relation V² + 4H² = 4F0², valid at every point on Earth's surface (with F0 being Earth's fixed equatorial field magnitude).
From the dipole field equations V=2F0 sinλ and H=F0 cosλ, squaring and combining gives V² + 4H² = 4F0² sin²λ + 4F0² cos²λ = 4F0²...
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