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tokens: in Fig. 7 . As can be seen, we now require 3 × 3 sets of MNn discs (i.e. 27 parameters) to obtain a fit at better than 10 per cent accuracy over 0.1 s ′ / r s 10. Figure 6. The twisted triaxial NFW density profile. The mass distribution is axisymmetric close to the centre and triaxial in the outskirts. Figure 7. MNn fit (dotted line) for a twisted triaxial NFW density profile (solid lines). We require 3 × 3 MNn discs for this more complex profile to get a good fit (better than 10 per cent accuracy over the range 0.1 s ′ / r s 10). 3.4 The potential and the force field So far, we have compared the original density profile with the one we obtain with our potential-density pair. It is interesting also to compare the potential generated by these density profiles and the force field as well. An important advantage of our potential-density pair is that the potential and the force field can be calculated analytically which is not the case, for instance, for the triaxial and twisted NFW profiles which have to be calculated numerically. We now compare the potential and the force field generated by a triaxial NFW density profile with those generated by our potential-density pair. We first calculate the potential generated by the original function, using the Chandrasekhar formula (Chandrasekhar Press 1969 ): This is an improper integral over the variable τ, where Once we have calculated the potential generated by the original density profile, now we calculate the potential generated by our superposition of MNn discs. This is just the sum of the potential generated by each single MNn disc that follows analytically from equation ( 3 ). We compare our MNn potential and force field with the true solution in Fig. 8 for the case of a triaxial NFW profile. Figure 8. Residuals of the potential and force field for a 3 × 2 MNn fit to the triaxial NFW density profile. 4 CONCLUSIONS We have presented a novel positive potential-density pair expansion for modelling galaxies, based on the MN disc. The key novelties of our method are: (i) we use three sets of such discs aligned along each symmetry axis, allowing us to model triaxial and twisted triaxial systems; (ii) we increase the efficiency of our expansion by allowing the scalelength parameter of each disc a to be negative; (iii) we introduce a constraint equation that ensures that the total density is everywhere positive; and (iv) we allow arbitrarily many discs in our expansion, allowing us to accurately model a wide range of density distributions relevant for modelling galaxies. We find that at better than 10 per cent accuracy in our density reconstruction, we can represent a radial and vertical exponential disc over 0.1–10 scalelengths/scaleheights with four MNn discs; an NFW profile over 0.1–10 scalelengths with four MNn discs; and a twisted triaxial NFW profile with three MNn discs per symmetry axis. Our potential-density pair is efficient, fully analytic, and well suited to reproducing the density