Chapter IV: Page

THE DIFFERENTIAL AND INTEGRAL CALCULUS 120

ITS TWO FUNDAMENTAL DIVISIONS 120

THEIR RELATIONS TO EACH OTHER 121
1. Use of the Differential Calculus as
preparatory to that of the Integral 123
2. Employment of the Differential
Calculus alone 125
3. Employment of the Integral Calculus
alone 125
Three Classes of Questions hence
resulting 126

THE DIFFERENTIAL CALCULUS 127
Two Cases: Explicit and Implicit Functions 127
Two sub-Cases: a single Variable or
several 129
Two other Cases: Functions separate or
combined 130
Reduction of all to the Differentiation of
the ten elementary Functions 131
Transformation of derived Functions for
new Variables 132
Different Orders of Differentiation 133
Analytical Applications 133

THE INTEGRAL CALCULUS 135
Its fundamental Division: Explicit and
Implicit Functions 135
Subdivisions: a single Variable or several 136
Calculus of partial Differences 137
Another Subdivision: different Orders of
Differentiation 138
Another equivalent Distinction 140
_Quadratures_ 142
Integration of Transcendental Functions 143
Integration by Parts 143
Integration of Algebraic Functions 143
Singular Solutions 144
Definite Integrals 146
Prospects of the Integral Calculus 148

CHAPTER V.

Page

THE CALCULUS OF VARIATIONS 151

PROBLEMS GIVING RISE TO IT 151
Ordinary Questions of Maxima and Minima 151
A new Class of Questions 152
Solid of least Resistance;
Brachystochrone; Isoperimeters 153

ANALYTICAL NATURE OF THESE QUESTIONS 154

METHODS OF THE OLDER GEOMETERS 155

METHOD OF LAGRANGE 156
Two Classes of Questions 157
1. Absolute Maxima and Minima 157
Equations of Limits 159
A more general Consideration 159
2. Relative Maxima and Minima 160
Other Applications of the Method of
Variations 162

ITS RELATIONS TO THE ORDINARY CALCULUS 163

CHAPTER VI.

THE CALCULUS OF FINITE DIFFERENCES 167

Its general Character 167
Its true Nature 168

GENERAL THEORY OF SERIES 170
Its Identity with this Calculus 172

PERIODIC OR DISCONTINUOUS FUNCTIONS 173

APPLICATIONS OF THIS CALCULUS 173
Series 173
Interpolation 173
Approximate Rectification, &c. 174

BOOK II.

GEOMETRY.