Cyclic order and dissection order of certain arcs

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Let arc A A in the conformal plane or on the sphere have local cyclic order three and cyclic order t t . It can be decomposed into a finite number of subarcs of cyclic order three. Let the disection order of A A be the minimum number of arcs in such a decomposition. The principal result of this paper is that the cyclic order t t and dissection order d d of A A satisfy the relation d + 2 ≦ t ≦ 3 d d + 2 \leqq t \leqq 3d . In establishing this result it is proved that a necessary and sufficient condition, that an arc of local cyclic order three shall be of global cyclic order three, is that there exists a circle meeting it only at the endpoints.