On Certain Crinkly Curves

Introduction.-In any field of geometric investigation the curves fall roughly into two classes, constituted respectively of the curves ordinarily investigated and of the other curves ; these unusual curves are in positive designation the crinkly curves.In this paper we are to investigate by interplaying graphic and analytic methods (in I) the continuous surface-filling oey-curves : x = cp (t) , y = yjr (t) : of Peano and Hilbert and (in II) the continuous tangentless yt-curve : y = yjr(t) : connected with Peano's curve.We define the various curves AT as point-forpoint limit-curves for n = oo of certain curves Kn (n = 1, 2 , 3 , • • •) ; these curves Kn are broken-line curves derivable each from the preceding by processes simple and such that the (nodal) extremities of the various w-links of Kt persist as corresponding points and also nodes of the Fl+l ; thus, the nodes of 7T are points of If; the set of all these nodes (for all n's) is on K everywhere dense.The curves AT are continuous and approach their point-for-point limit-curve K uniformly ; AT is accordingly continuous, a conclusion however which is geometrically evident.From the continuity of K and the presence of the set of nodes the properties of K follow in such a way as to appeal vividly to the geometric imagination.Indeed the yt-cuvve from the simplicity of its geometric definition and from the intuitive clearness of its properties appears to be fit to replace the classical Weierstrass curve as the standard example of continuous curves having no tangents, since, further, we develop closer knowledge of its progressiveand regressive-tangential properties (II § § 8, 11).The basal notions of this paper were communicated to Chicago colleagues in February and March, 1899.-Part II has certain relations of content with the interesting paper by Steinitz, Stetigkeit und Differentialquotienten, Mathematische