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In this paper it is found that a three-quadrupole focussing system can be morphed continuously through FFD, FDF and DFF variants and back again while maintaining stable optics and even keeping the two transverse tunes constant. This relates to non-scaling FFAGs, where the magnet gradients define both the focussing and the variation of the field with momentum as the closed orbit sweeps across it. A two-lens focussing system cannot change the sign of either gradient without becoming unstable, meaning non-scaling FFAGs built with such a lattice eventually encounter too large a magnetic field at low energies. However, a theoretical system of magnet field variations using three lenses, with a potentially unlimited energy range and fixed tunes is presented here.
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