| CHAP. I.INTRODUCTION Plane curves. Invariant relations. Existence theorems for implicit functions. Algebraic curves. II.THE ELEMENTARY PROPERTIES OF TANGENTS AND NORMALS . Tangents. Normals. Arcs. Differentials. Conventions of sign. Limiting ratios of arcs, chords and tangents. Points ofinflexion. Convexity and concavity. III.THE CURVATURE OF PLANE CURVES 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 diffrentials. Frenet's for- mulae for twisted curves. Evolutes and involutes. IV.THE THEORY OF CONTACT Distance of a curve from a point near it. Contact of order 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 . 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 Form of f(x, 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 Definition and fundamental properties. Asymptotes as limits oftangentsand chords. Asymptotes of algebraic curves. Rules for rectilinear asymptotes. Parabolic and curvilinear asymp- totes. |
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