Invariants of differential configurations in the plane

Abstract

Suppose we have n points (x,, y,,), (v = 1, 2, 3, ***, n), in mutually general positions,t and let us assume that through each point (x,,, y,) there pass A, regular analytic elements of order r. The elements belonging to any point may have contact of zero order-that is, merely intersect-or some of them may have contact with each other of any order less than r. The totality of the elements considered will be called a differential configuration of order r. If such a differential configuration be subjected to a group of point transformations, it will be changed always into another of the same type, that is, having the same constants n, X,,, and the order of contact of any two of the elements preserved. The numbers n, X1, are arithmetic invariants of the configuration under the group of point transformations, but will not necessarily be so under a group of contact transformations. Rabutt appears to have been the first to pay systematic attention to the differential invariants of such configurations. He considered two groups-the group of all point transformations in the plane, and that of all contact transformations in the plane. He claimed for his method that it gave not only all configurations having invariants-both relative and absolute-but also all possible types of such invariants. Doubt was thrown on the generality of this method by Kasner, ? who found (in another connection) a type of invariant which Rabut had declared impossible, and discussed a geometric interpretation of the samne. (Rabut's actual error is pointed out by the writer in the present paper.)