where i a =-(2(ai)2)1/2 and where F satisfies the sole condition that F -O 0 as a -O 0, b 0. The coordinate system is right-regular if I b I replaces I a I in (1.1). A coordinate system a, *., at is analytic if the coordinates (ab)t of ab are expressible as power series in a', * , a, bl, * , bT which converge for some domain: I a I < 6, I b I < 6. A local group G,. is a local Lie group if it is locally isomorphic to a Gr with an analytic coordinate system. The purpose of this note is to prove the following theorem. (1.2) THEOREM. A necessary and sufficient condition that an r-parameter local group be a local Lie group is that it admit a left(or right-) regular coordinate system. It is easy to see that an analytic coordinate system for G6 is leftand rightregular. We have therefore only to show that a Gr which admits left-regular coordinates admits analytic coordinates. The proof will depend on certain constructions and formulas in our paper [1] in which the composition functions (ab)' were assumed to possess continuous derivatives with respect to a', * * *, a' and to satisfy Lipschitz conditions with respect to bl, * * *, bV. Enough will be repeated here to make frequent reference to [1] unnecessary. Left-regularity seems to be the weakest condition yet considered for r-parameter local groups which leads to analyticity.2 The composition function ab defining a local group Gr is of course assumed to satisfy the group axioms in a neighborhood of the identity. It is interesting to note however that in addition to continuity, only associativity and the existence of an identity need be assumed for ab. The proof of the existence of continuous inverses runs along the lines of an argument given in [2].