1. By an oval we mean a convex closed curve. For the sake of simplicity the following discussion will be restricted to curves whose curvature is continuous and never vanishes, although our results can be extended, with suitable modifications, to the general case. An extensive use will be made of vector-calculus notation, the bold-faced types designating vectors. Thus the position-vector r of a variable point on the curve will have continuous second derivative with respect to s, length of arc. The symbols t and a will designate respectively the unit tangent and unit normal vectors whose orientation coincides with that of the OX, 0 Y axes. We shall consider simultaneously a pair of opposite points (P, P'), at which the tangents are parallel. The distance between these tangents and the chord PPi will be called respectively the width of the curve, and the diameter at P, and will be designated by p& and d. The width It, as well as the curvature x, and the radius of curvature e1/x, will be assumed >0. Finally if a point P' is opposite to P, the geometric quantities which correspond to P' will be designated by the same letters as those for P, with addition of a prime ('). We assume