Radius of curvature with rectilinear and polar co-ordinates. Change of the independent variable.
Contacts of different orders of plane curves. Osculating curves of a given kind. Osculating straight line. Osculating circle. It is identical with the circle of curvature.
Application of the method of infinitesimals to the determination of the radius of curvature of certain curves geometrically defined. Ellipse, cycloid, epicycloid, &c.
Evolutes of plane curves. Value of the arc of the evolute. Equation to the involute of a curve. Application to the circle. Evolutes considered as envelops. On envelops in general. Application to caustics.
Lessons 24–27. Geometrical Applications continued. Curvature of Lines of Double Curvature and of Surfaces.
Osculating plane of a curve of double curvature. It may be considered as passing through three points infinitely near to one another, or as drawn through a tangent parallel to the tangent infinitely near to the former. Center and radius of curvature of a curve of double curvature. Osculating circle. Application to the helix.
Radii of curvature of normal s of a surface. Maximum and minimum radii. Relations between these and that of any , normal or oblique.
Use of the indicatrix for the demonstration of the preceding results. Conjugate tangents. Definition of the lines of curvature. Lines of curvature of certain simple surfaces. Surface of revolution. Developable surfaces. Differential equation of lines of curvature in general.
Lesson 28. Cylindrical, Conical, Conoidal surfaces, and Surfaces of Revolution.