VI. Evaluation of Functions which for certain values of the

Variable become indeterminate

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

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

Equations from their Geometrical Properties

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

X. Singular Points in Curves

XI.

On the Tracing of Curves from their Equations...

162

175

188

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

XIII. Application of the Differential Calculus to Geometry of

Three Dimensions

XIV. Envelops to Lines and Surfaces

XV. General Theorems in the Differential Calculus.

200

224

237

IX.

Quadrature of Areas and Surfaces, Rectification of Curves,

and Cubature of Solids

412

DIFFERENTIAL CALCULUS.

CHAPTER I.

DIFFERENTIATION.

Functions of One Variable.

IF 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

y being some function of x, and u some function of y. This theorem may be extended to any number of functions, so that

du

dx

(7)

u = log"; which signifies not the nth

power of

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

1

a log a log2 x ...... log"-1 x

du

dx x

a