Abstract
In a series of papers [1; 2; 3; 4], K. T. Chen introduced and studied certain infinite series of numbers associated with paths in Euclidean n-space. These numbers were invariants under translations, and in [4] he proved that they uniquely characterize paths under translations. This paper's purpose is investigating similar path invariants for other transformation groups. Integral invariants are studied using prolongations in the context of C. Ehresmann's jets [5]. A general discussion of the ideas involved is given in ?1. ?2 is devoted to differential equations and some illuminating examples. We find in ?3 a sufficient condition that a group's invariants uniquely characterize paths. The theory also applies to pseudo groups having sufficiently regular behavior. All structures assumed C' (infinitely differentiable). 1. INVARIANTS
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