The behavior of the characteristics2 of a vector field on the two-sphere has been extensively studied in classical papers of Poincar6 and Bendixson.3 The purpose of this note will be to add a few comments to their theory. In particular two formulas will be given relating the numbers of closed characteristics of various types with the numbers of critical points of various types. A convenient presentation of the Poincar6-Bendixson theory has been given by Lefschetz3, and this may be used as a basic reference. Let R be a region of the two-sphere; and consider a vector field in R which is sufficiently regular so that exactly one characteristic passes through each noncritical point. The following assumptions will be made: (1) Each critical point is either a node, focus, or saddle. (See Lefschetz, pp. 120-124. The node and focus are topologically the same.) (2) R has at most a finite number of boundaries, and each boundary is a differentiable curve which is nowhere tangent to the vector field. (3) There are at most a finite number of closed characteristics. It follows from these assumptions that there are at most a finite number of critical points. By a closed characteristic polygon will be meant either a closed characteristic or a sequence c0, cl, , cn = co of compatibly oriented characteristics such that the end point of each ci is the beginning point of ci+1 . The vertices of such a polygon must be saddle points. It is easily proved that there are at most a finite number of closed characteristic polygons. By a singular element will be meant either: (1) a critical point, (2) a boundary curve, or (3) a closed characteristic polygon, together with a preferred side. By a source will be meant a singular element which is unstable in the sense that all characteristics in some neighborhood tend away from the element. Thus a source is either: (1) an unstable node or focus, (2) a boundary curve along which the vector field points into R, or (3) a closed characteristic polygon together with a preferred side, such that the characteristics on this side spiral away from the polygon. Similarly a sink will mean a singular element which is stable in the sense that all characteristics in some neighborhood tend towards the element. For each singular element s define the function ?(s) as follows. Let e(s) equal