On the different orders of infinitely small quantities. Infinitely small quantities of a certain order may be neglected in respect of those of an inferior order. The infinitely small increment of a function is in general of the same order as the corresponding increment of the variable, that is to say, their ratio has a finite limit. Definitions of the derivative and differential of a function of a single variable. Tangents and normals to plane curves, whose equation in linear or polar coordinates is given. A function is increasing or decreasing, according as its derivative is positive or negative. If the derivative is zero for all values of the variable, the function is constant. Concavity and convexity of curves; points of inflection. Principle of function of functions. Differentiation of inverse functions. Differentials of the sums, products, quotients, and powers of functions, whose differentials are known. General theorem for the differentiation of functions composed of several functions. Differentials of exponential and logarithmic functions. Differentials of direct and inverse circular functions. Differentiation of implicit functions. Tangents to curves of double curvature. Normal plane. Differential of the area and arc of a plane curve, in terms of rectilinear and polar co-ordinates. Differential of the arc of a curve of double curvature. Applications to the cycloid, the spiral of Archimedes, the logarithmic spiral, the curve whose normal, sub-normal, or tangent, is constant; the curve whose normal passes through a fixed point; the curve whose arc is proportional to the angle which it subtends at a given point. Derivatives and differentials of different orders of functions of one variable. Notation adopted. Remarks upon the singular points of plane curves. LESSONS 10-13. _Derivatives and Differentials of Functions of Several Variables._ Partial derivatives and differentials of functions of several variables. The order in which two or any number of differentiations is effected does not influence the result. Total differentials. Symbolical formula for representing the total differential of the _n_^{th} order of a function of several independent variables. Total differentials of different orders of a function; several dependent variables. Case where these variables are linear functions of the independent variables. The infinitesimal increment of a function of several variables may in general be regarded as a linear function of the increments assigned to the variables. Exceptional cases. Tangent and normal planes to curved surfaces. LESSONS 14-18. _Analytical Applications of the Differential Calculus._ Development of F(_x + h_,) according to ascending powers of _h_. Limits within which the remainder is confined on stopping at any assigned power of _h_.