  For physical theories, the degree of arbitrariness of a system is of great
importance, and is often closely linked to the concept of degree of freedom,
and for most systems this number is far from obvious. In this paper we present
an easy to apply algorithm for calculating the degree of arbitrariness for
large classes of systems formulated in the language of moving frames, by mere
manipulation of the indices of the differential invariants. We then give
several examples illustrating our procedure, including a derivation of the
degree of arbitrariness of solutions of the Einstein equation in arbitrary
dimensions, which is vastly simpler than previous calculations, the degree of
arbitrariness of gauge theories of arbitrary groups under Yang-Mills type
equations, for which until now only specific cases have been calculated by
rather messy means, and the degree of arbitrariness of relativistic rigid flow
with or without additional constraints, which has not been derived before.
Finally we give the proof of our algorithm.
