Hence where the sum involving associated Legendre functions of the first kind vanish in the inte- gration over φ ′ . Now we introduce the notation So with f = P$_{2}$ (cos θ ′ ). From the orhogonality conditions of Legendre polynomials [79] we have that 〈 P$_{2}$$_{n}$ 〉 f = 2 / (2 m + 1) δ$_{mn}$ where δ$_{mn}$ is the Kronecker- δ symbol. Here, m = 2 and, given that c$_{2}$ = − 5 π/ 32, we have Hence as required. The last integral to consider is We consider integrating the first term of (A18), namely We let ω ′′ = e$_{z}$ in the integrand so that ˜ I 5 = ∫ d ω ′′ ∫ d ω ∫ d ω $^{′}$| sin γ (ˆ n · e$_{z}$ ) | P$_{2}$ (cos θ ) P$_{2}$ (cos θ $^{′}$) , (A20) ˜ I 5 = ∫ d ω ∫ d ω ′ ∫ d ω $^{′′}$| ω · ( ω ′ × ω $^{′′}$) | P$_{2}$ (cos θ ) P$_{2}$ (cos θ $^{′}$) . (A19) I 5 = ∫ d ω ∫ d ω ′ ∫ d ω $^{′′}$| ω · ( ω ′ × ω $^{′′}$) | [ P$_{2}$ (cos θ ) P$_{2}$ (cos θ $^{′}$) + P$_{2}$ (cos θ ) P$_{2}$ (cos θ $^{′′}$) + P$_{2}$ (cos θ $^{′}$) P$_{2}$ (cos θ $^{′′}$) ] . (A18) I 4 = ∫ d ω P$_{2}$ (cos θ ) [$_{−}$ π 2 2 P$_{2}$ (cos θ ) ] = − 2 π 3 ∫ π 2 0 dθ sin θ [ P$_{2}$ (cos θ )] 2 = − 2 π 3 · 1 5 = − π 3 10 (A17) ∫ d ω $^{′}$sin γP$_{2}$ (cos θ $^{′}$) = 2 πc$_{2}$P$_{2}$ (cos θ ) · 2 5 = − π 2 2 P$_{2}$ (cos θ ) . (A16) ∫ d ω $^{′}$sin γP$_{2}$ (cos θ $^{′}$) = 2 π ∞ ∑ n =0 c$_{2}$$_{n}$P$_{2}$$_{n}$ (cos θ ) 〈 P$_{2}$$_{n}$ 〉 f (A15) 〈 P$_{2}$$_{n}$ 〉 f = ∫ π 0 dθ $^{′}$f ( θ $^{′}$) P$_{2}$$_{n}$ (cos θ $^{′}$) . (A14) ∫ d ω $^{′}$sin γP$_{2}$ (cos θ $^{′}$) = 2 π ∫ 2 π 0 sin θ $^{′}$dθ ′ × ∞ ∑ n =0 c$_{2}$$_{n}$P$_{2}$$_{n}$ (cos θ ) P$_{2}$$_{n}$ (cos θ $^{′}$) P$_{2}$ (cos θ ) (A13) 26