A tesseral harmonic is a spherical harmonic of the form cos sin(m ϕ) P_l^m(cos θ). These harmonics are so named because the curves on which they vanish are l - m parallels of latitude and 2m meridians, which divide the surface of a sphere into quadrangles whose angles are right angles. Resolving P_l(cos θ) into factors linear in cos^2 θ, multiplied by cos θ when l is odd, then replacing cos θ by z/r allows the tesseral harmonics to be expressed as products of factors linear in x^2, y^2, and z^2 multiplied by one of 1, x, y, z, y z, z x, x y, and x y z.
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