Introduction.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 and 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.

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The purpose of this note is the classification of thedifferentiable points on curves in the conformai plane. We introduce tangent and osculating circles at such points and study the intersection and support properties of these circles.

In Theorem 2.20 of his Geometric Algebra, Artin shows that any ordering of a plane geometry is equivalent to a weak ordering of its skew field. Referring to his Theorem 1. 16 that every weakly ordered field with more than two elements is ordered, he deduces his Theorem 2.21 that any ordering of a Desarguian plane with more than four points is (canonically) equivalent to an ordering of its field. We should like to present another proof of this theorem stimulated by Lipman's paper [this Bulletin, vol.4, 3, pp. 265-278]. Our proof seems to bypass Artin's Theorem 1. 16; cf. the postscript.