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Remembering that in our coordinate system ω = (sin θ, 0 , cos θ ) and ω ′ = (cos θ ′ sin θ ′ , sin φ ′ sin θ ′ , cos θ ′ ), 

<img src='content_image/339250.jpg'>

as required. (Here the inner two integrals alone give π $^{3}$/ 2 which can be found straight- forwardly). There are three further integrals required for the bifurcation analysis. I 3 = ∫ d ω [ P$_{2}$ (cos θ )] 2 = 4 π/ 5 trivially. Let us turn our attention to 

We expand sin γ as 

with coefficients 

where the double factorial is defined by 

Therefore the inner integral of (A7) becomes 

<img src='content_image/339247.jpg'>

<img src='content_image/339246.jpg'>

<img src='content_image/339245.jpg'>

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<img src='content_image/339244.jpg'>

We now utilise the addition formula for Legendre polynomials, 

<img src='content_image/339243.jpg'>

