The paper considers a mechanical system subjected to the action of ideal non-holonomic connections. It starts from the ga-th order principle, which, using the Gaussian function of the least constraint, is transformed into the form (1). It is further shown that the known forms (3) and (4) are equivalent to (1). In the case that the mechanical system is subjected to the action of ideal non-holonomic connections of higher order, the differential principle (1) can be written in the form (5), from which it is obvious that the Gauss function of the least constraint allows the equations of motion of the considered non-holonomic system to be expressed in in the most condensed form (6). At the end, a review of the solution to this problem was given in works [3] and [4]. It has been shown that those works contain certain inaccuracies.