The two conformal invariants of fifth order

Abstract

In the problem of classifying curvilinear angles in the plane with respect to the group of conformal transformations, the simplest invariant (other than angular magnitude 0) which occurs is found in the case of a horn angle of first order contact (formula (1)). This is a differential invariant of third order, since it involves third derivatives of the two curves composing the angle. There are no invariants of fourth order (nor of any even order), but there are known to be just two of fifth order, arising in the cases of the horn angle of second order contact and the general right angle, respectively.t The main object of this paper is to determine the two invariants of fifth order explicitly. The results appear in formulas (15) and (23) below.t It is shown elsewhere? how the latter of these expressions can be derived from the former through conformal symmetry. As a corollary we obtain in ?3 the invariant of lowest order of a single curve under the inversion group, first found by G. W. Mullins. Generalizations to the case of angles drawn on a curved surface have been given by G. Comenetz.11 I wish to acknowledge the valuable assistance of Miss A. Vassell and J. De Cicco in calculating the invariants, and of G. Comenetz in connection with ?3 and the use of oriented curves. 1. Horn angle of second order. We use the term curvilinear angle for the figure formed by an ordered pair of oriented, analytic curve-elements having a common point 0, the vertex of the angle. The symbol-y is employed for the curvature of a curve, -y' for the derivative dy/ds with respect to arc-length s, oy for the second derivative d2y/ds2, and so on. Given a curvilinear angle, the symbols, -y', 4y', * denote the curvature and its successive arc-length derivatives for the first side of the angle, evaluated at the vertex 0; and -y2, 42, 42'), ... are the corresponding