Interesting future work could involve examining the phase behaviour of binary mixtures of polarizable platelets, that is particles that interact with some external applied magnetic field, as has been investigated for binary rod mixtures [68, 69]. Also, binary mixtures of rods and platelets could be investigated [70–74]. Another line of investigation would be to incorporate polydispersity into the model since many platelet systems often have a significant polydispersity, see for example [1]. This would lead to an extension of the work by Speranza and Sollich and others on rod-like particles, [75–77] and by Wensink and Vroege on thickness-polydisperse platelets [78]. Here we evaluate some integrals that are important in the bifurcation analysis. The first, I $_{1}$, is a standard integral in excluded volume calculations Without loss of generality, let ω ′ = e$_{z}$ in the integrand, where e$_{z}$ is the unit vector in the z -direction (the symmetry axis for the system). Now we have Remembering that in our coordinate system ω ′ = (cos θ ′ sin θ ′ , sin φ ′ sin θ ′ , cos θ ′ ), as required. The second integral arises in FMT. We wish to integrate the triple scalar product over three unit orientation vectors, ω , ω ′ , ω ′′ . Similar to the I 1 calculation, let ω ′′ = e$_{z}$ in the integrand. I 1 = ∫ d ω ′ ∫ d ω √$_{1}$ − cos 2 θ = ∫ d ω ′ ∫ 2 π 0 dφ ∫ π 0 dθ sin 2 θ = π 2 ∫ d ω ′ = 4 π 3 (A3) I 1 = ∫ d ω ′ ∫ d ω √ 1 − ( ω · e$_{z}$ ) $^{2}$. (A2) I 2 = ∫ d ω ∫ d ω ′ ∫ d ω $^{′′}$| ω · ( ω ′ × ω $^{′′}$) | . (A4) I 2 = ∫ d ω ′′ ∫ d ω ∫ d ω $^{′}$| ω · ( ω ′ × e$_{z}$ ) | . (A5) 24 Appendix A: Calculation of Relevant Integrals I 1 = ∫ d ω ∫ d ω $^{′}$sin γ = ∫ d ω ∫ d ω ′ √ 1 − ( ω · ω ′ ) $^{2}$. (A1)