ilmm mnm IINBINQ LIST QECa 1921 Digitized by the Internet Archive in 2007 with funding from Microsoft Corporation http://www.archive.org/details/elennentarydifferOOfowluoft Cambridge Tracts in Mathematics "; and Mathematical Physics General Editors J. G. LEATHEM, Sc.D. G. H. HARDY, M.A., F.R.S. No. 20 The Elementary Differential Geometry of Plane Curves CAMBRIDGE UNIVERSITY PRESS C. F. CLAY, Manager LONDON : FETTER LANE, E.G. 4 NEW YORK : G. P. PUTNAM'S SONS BOMBAY ") CALCUTTA I MACM I LLAN AND CO., Ltd. MADRAS J TORONTO : J. M. DENT AND SONS, Ltd. TOKYO : MARUZEN-KABUSHIKI-KAISHA ALL RIGHTS RESERVED THE <> ELEMENTARY DIFFERENTIAL GEOMETRY OF PLANE CURVES BY R. H. FOWLER, M.A, Fellow of Trinity College, Cambridge CAMBRIDGE AT THE UNIVERSITY PRESS 1920 4 PKEFACE THIS tract is intended to present a precise account of the elementary differential properties of plane curves. The matter contained is in no sense new, but a suitable connected treatment in the English language has not been available. As a result, a number of interesting misconceptions are current in English text books. It is sufficient to mention two somewhat striking examples, (a) According to the ordinary definition of an envelope, as the locus of the limits of points of intersection of neighbouring curves, a curve is not the envelope of its circles of curvature, for neighbouring circles of curvature do not intersect, (b) The definitions of an asymptote—(1) a straight line, the distance from which of a point on limit of the curve tends to zero as the point tends to infinity ; (2) the a tangent to the curve, whose point of contact tends to infinity—are not equivalent. The curve may have an asymptote according to the former definition, and the tangent may exist at every point, but have no limit as its point of contact tends to infinity. The subjects dealt with, and the general method of treatment, are similar to those of the usual chapters on geometry in any Cours d'Analyse, except that in general plane curves alone are considered. At the same time extensions to three dimensions are made in a somewhat arbitrary selection of places, where the extension is immediate, and forms a natural commentary on the two dimensional work, or presents special points of interest (Frenet's formulae). To make such extensions systematically would make the tract too long. The subject matter being wholly classical, no attempt has been made to give full references to sources of information ; the reader however is referred at most stages to the analogous treatment of the subject in the Cours or Traite d'Analyse of de la Valine Poussin, Goursat, Jordan or Picard, works to which the author is much indebted. VI PREFACE In general the functions, which define the curves under considera- tion, are (as usual) assumed to have as many continuous differential coefficients as may be mentioned. In places, however, more particularly at the beginning, this rule is deliberately departed from, and the greatest generality is sought for in the enunciation of any theorem. The determination of the necessary and sufficient conditions for the truth of any theorem is then the primary consideration. In the proofs of the elementary theorems, where this procedure is adopted, it is believed that this treatment will be found little more laborious than any rigorous treatment, and that it provides a connecting link between Analysis and more complicated geometrical theorems, in which insistence on the precise necessary conditions becomes tedious and out of place, and suitable sufficient conditions can always be tacitly assumed. At an earlier stage the more precise formulation of conditions may be regarded as (1) an important grounding for the student of Geometry, and (2) useful practice for the student of Analysis. The introductory chapter is a collection of somewhat disconnected theorems which are required for reference. The reader can omit it, and to refer to it as it becomes necessary for the understanding of later chapters. I wish to express my great indebtedness to the Editor, Mr G. H. Hardy, and also to Mr J. E. Littlewood and Dr T. J. I' A. Bromwich, for assistance and advice in the preparation of this tract. R. H. F. Octoh&r 1919. <:> CONTENTS CHAP. PAGE I. Introduction 1 Plane curves. Invariant relations. Existence theorems for implicit functions. Algebraic curves. II. The elementary properties of tangents and normals . 8 Tangents. Normals. Arcs. Differentials. Conventions of sign. Limiting ratios of arcs, chords and tangents. Points of inflexion. Convexity and concavity. III. The curvature of plane curves 24 Curvature. Circle, centre, and radius of curvature. Geometrical properties of the centre of curvature. Order of approximations to the circle of curvature. Newton's method. Direction cosines of tangent and normal, and their dififerentials. Frenet's for- mulae for twisted curves. Evolutes and involutes. IV. The theory of contact 45 Distance of a curve from a point near it. Contact of order n for two curves. Osculating curves. Contact of surfaces, and surface and curve. Osculating surfaces. Contact of twisted curves. Osculating twisted curves. V. The theory of envelopes 58 Characteristic points. Definition of envelopes. The envelope as the limit of intersections of neighbouring curves. Contact properties of the envelope. Isolated points of exceptionally high order contact. Envelopes with contact everywhere of high order. Families of circles. Similar problems in three dimensions. VI. Singular points of plane curves 80 Form of / (.r, y) near a singular point. Nature of the curves so defined. Branches. Possible tangents of branches at a singular point. Singular points of the second order. Isolated points, double points and cusps. Singular points of order n. VII. Asymptotes of plane curves 89 Definition and fundamental properties. Asymptotes as limits of tangents and chords. Asymptotes of algebraic curves. Rules for rectilinear asymptotes. Parabolic and curvilinear asymp- totes. i u fi If; . : CHAPTER I INTRODUCTION § 110. We assume in this tract that the reader is acquainted with the ordinary elementary theorems of the differential and integral calculus, as developed, for example, in Hardy's Pui^e Mathematics (2nd Edition, 1914); we apply these theorems to the geometry of plane curves. We shall require more than is there given concerning implicit functions, especially algebraic functions and the curves defined thereby. Such theorems of this type as we require frequently are quoted with references in § 1*50. The more important special properties of algebraic functions are summarized in § 1*60. We shall use freely the symbols ^, 0, o, whose use is now classical, and occasionally ^ and >. The reader who is not acquainted with any of them will find the meaning of f^, 0, ^ and >= explained in Hardy's tract 'Orders of Infinity' {Cambridge Mathematical Tracts, No. 12). The definition of o is as follows If fix) and g (x) are any functions of x, and g (x) is positive* fm^ all sufficiently large values of x, we write f{x) = o{g{x)), ivhen \f{x) {x) -> \ jg as x—¥ cc A similar definition applies when x tends to zero, or any other finite limit, instead of to infinity. The introduction of o into Analysis is due to Landau, vide Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Vol. i, p. 59. The symmetry of the differential notation and the use of direction cosines are of vital importance in three-dimensional geometry. They can be used with advantage in two dimensions and lend themselves at once to the necessary generalizations. They are therefore used freely here. It is, however, important that the reader should realise the * Alternatively, it will be convenient for our purposes to allow g {x) to be negative instead of positive in the above definition, and also in the definition of O. The only essential requisite in these definitions is that g [x) should not vanish for large values of X. F. 1 2 INTRODUCTION precise uature of the statement made by a differential formula, and this is frequently emphasised. A limited selection of examples is given at the ends of the chapters. Besides their more obvious function, these are intended to provide a summary of some of the more important extensions of the -theorems proved in the text. References or sketches of a proof are therefore given in such cases, which should enable the reader to complete the proofs. § 1'20. Plane curves. We regard a plane curve as the locus of points satisfying the equations for a given range of values of t^tQ^t^t^, say) for which <f>i (t), <f>2(t) are continuous single-valued functions of t. A point F on the curve is regarded as identified with a value of t. The variable t is real, and oj and y are also always real. We consider throughout only real points and curves. More information than this about <^i (t) and </)2 (t) will always be required, the amount varying from problem to problem. We may specify conditions to be satisfied by <^i (t) and 02 (0 either (1) at a point P, i.e. when t = t„, or (2) in the neighbourhood of or "near" a point P, i.e. in the neighbourhood of ^oj or (3) throughout the interval FQ, i.e. when t^'^t^ti. We say that the point F (t) lies between the points Qi (ti) and QaCO 0^^ *^® curve, when ti<t <t2', also that the point Q{t) tends to F(to), or Q—>F, F and Q being points on the curve, when t—>to. A particular case of great importance occurs when x-t on y = t, and the curve is given in one of the forms y=<f>^(a:), x = ^^{y).