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. In [3], the authors discussed a definition of the conformal differentiability of an arc at a point in the conformal plane. It was based on the postulation of tangent circles and of an osculating circle. The intersection an'd support properties of all the circles through a differentiable point were studied, and by means of these properties the differentiable points were classified into various types. Each type was uniquely described by a certain triple of numbers, the characteristic of that type. In this paper, relationships between the characteristic of a differentiable point and its cyclic order are established. In this connection some-partly familiar-differentiability properties of arcs of order three will be discussed. Our main results are stated in five theorems; cf. ??2.1, 3.4, 3.5, 5.1, 5.9.

Characteristic and order of differentiable points in the conformal plane