To find the relations between the moments of inertia about different axes through any assigned point O, we take O as origin. Since the square of the distance of a point (x, y, z) from the axis

x= y= z
λ μν

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is x2 + y2 + z2 − (λx + μy + νz)2, the moment of inertia about this axis is

I = Σ [m { (λ2 + μ2 + ν2) (x2 + y2 + z2) − (λx + μy + νz)2} ]
= Aλ2 + Bμ2 + Cν2 − 2Fμν − 2Gνλ − 2Hλμ,

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provided

A = Σ {m (y2 + z2)},   B = Σ {m (z2 + x2)},   C = Σ {m (x2 + y2)},
F = Σ (myz),   G = Σ (mzx),   H = Σ (mxy);

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i.e. A, B, C are the moments of inertia about the co-ordinate axes, and F, G, H are the products of inertia with respect to the pairs of co-ordinate planes. If we construct the quadric