If O is an ordinary point of a plane curve and the tangent and inward normal at O are taken as axes, then, under certain conditions, the coordinates x and y of a neighbouring point P on the curve can be expanded in powers of s, the arc OP, or of Ψ, the angle between the tangents at O and P. Lamb(3) gives these expansions as far as the terms in s4 and Ψ3. Dockeray(1) goes as far as the terms in s6, but there seem to be three errors in his results. No one, I believe, has given the general terms. I obtain these for both pairs of expansions. Fowler(2) gives the term in s2r+1 in the expansion of x in the special case where, owing to the vanishing of the curvature and of its first r − 2 derivatives, the expansion of y begins with the term in sr+1. I shall give reasons for disagreeing with this result.