Development of F(_x_,) according to powers of _x_ or _x - a_; _a_ being a quantity arbitrarily assumed. Application to the functions sin(_x_,) cos _x_, _a^{x}_, (1 + _x^{m}_) and log.(1 + _x_.) Numerical applications. Representation of cos _x_ and sin _x_ by imaginary exponential quantities. Developments of cos^{m} _x_ and sin^{m} _x_ in terms of sines and curves of multiples of _x_. Development of F(_x + h, y + k_,) according to powers of _h_ and _k_. Development of F(_x, y_) according to powers of _x_ and _y_. Expression for the remainder. Theorem on homogeneous functions. Maxima and minima of functions of a single variable; of functions of several variables, whether independent or connected by given equations. How to discriminate between maxima and minima values in the case of one and two independent variables. True values of functions, which upon a particular supposition assume one or another of the forms 0/0, ∞/∞, ∞ + 0, 0^0, 4^∞ LESSONS 19-23. _Geometrical Applications. Curvature of Plane Curves._ Definition of the curvature of a plane curve at any point. Circle of curvature. Center of curvature. This center is the point where two infinitely near normals meet. 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 sections of a surface. Maximum and minimum radii. Relations between these and that of any section, 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._ Equations of these surfaces in finite terms. Differential equations of the same deduced from their characteristic geometrical properties. INTEGRAL CALCULUS.