V.

Application of the Differential Calculus to the Development

of Functions

52

VI.

Evaluation of Functions which for certain values of the

Variable become indeterminate

79

VII. Maxima and Minima.......

94

VIII. On the Generation of Curves and the Investigation of their

Equations from their Geometrical Properties

129

IX.

X.

On the Tangents, Normals, and Asymptotes to Curves...... 144

Singular Points in Curves

162

XI.

On the Tracing of Curves from their Equations ......

175

XII. On the Curvature of Curved Lines.........

188

XIII. Application of the Differential Calculus to Geometry of

Three Dimensions

200

XIV. Envelops to Lines and Surfaces

224

XV. General Theorems in the Differential Calculus.......

237

IX. Quadrature of Areas and Surfaces, Rectification of Curves,

and Cubature of Solids

412

DIFFERENTIAL CALCULUS.

CHAPTER I.

DIFFERENTIATION.

Functions of One Variable.

Ir u be an explicit function of x, which is of a complicated form, it may generally be reduced to the differentiation of simpler functions by means of the theorem

du du dy

=

dx dy dx'

y being some function of x,

and u some function of y.

This theorem may be extended to any number of functions,

(7)

dx x log x

u = log"; which signifies not the nth power of

the logarithm of x, but the nth logarithm of that quantity,