Introduction. The classical reduction theory of canonical systems with n degrees of freedom assumes that there is known a function group of first integrals.! The reduction of the degree of freedom is then carried out by using the Hamilton-Jacobi theory or an equivalent approach. In the present paper a more general problem will be considered, since it will not be assumed that the known functions, or hypersurfaces in the phase-space, are represented by first integrals. In fact, the first

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... will be replaced by invariant relations, % so that, in particular, the known hypersurfaces need not form a continuous family, but may be isolated. It should be mentioned that the generalization of the reduction problem to the case where the given first integrals are replaced by invariant relations is not an artificial problem but one which arises quite naturally in the simplest applications. If, for instance, one wants to reduce the degree of freedom of the problem of three bodies by means of the classical first integrals in an explicit and symmetrical manner, one is compelled to replace these first integrals by certain of their combinations which form a complicated system of invariant relations and not a system of first integrals. § The treatment of the general case of invariant relations will differ from the usual treatment of the case of first integrals in that all considerations will be based on a parametrization of the system of invariant relations, this parametrization being symmetrical with respect to the n coordinates and n impulses. Needless to say, the usual treatment of first integrals, based on the Hamilton-Jacobi theory, cannot be applied in the general case under consideration. In the particular case where the system of invariant relations is a system of first integrals ( §10), the treatment of the general case goes over into a