  A generalisation of Chaplygin's Reducing Multiplier Theorem is given by
providing sufficient conditions for the Hamiltonisation of Chaplygin
nonholonomic systems with an arbitrary number $r$ of degrees of freedom via
Chaplygin's multiplier method. The crucial point in the construction is to add
an hypothesis of geometric nature that controls the interplay between the
kinetic energy metric and the non-integrability of the constraint distribution.
Such hypothesis can be systematically examined in concrete examples, and is
automatically satisfied in the case $r= 2$ encountered in the original
formulation of Chaplygin's theorem. Our results are applied to prove the
Hamiltonisation of a multi-dimensional generalisation of the problem of a
symmetric rigid body with a flat face that rolls without slipping or spinning
over a sphere.
