MATHEMA TICS: A. D. MICHAL THE GROUP MANIFOLD OF FINITE CONTINUOUS POINT AND FUNCTIONAL TRANSFORMATION GROUPS By ARIsTomII

has recently been studied by Cartan' with the aid of ingenious geometric methods peculiar to his way of looking at an affinely connected manifold.2 From the analytic and invariant-theoretic standpoint, however, Cartan's geometric methods are by no means the last word on the subject. Furthermore, the proof of many of his theorems bring into play the variables xl, x2, .. ., x". It is the purpose of this paper to develop some of the essential results of Cartan's theory analytically in such a way as to make the theory applicable to finite continuous groups of functional transformations in function space as well as to point transformations in ndimensional space, and to exhibit various fundamental tensor invariants of the manifold in an arbitrary co6rdinate system. A simple example of an r-parameter group of functional transformations is given by