CHAPTER IV.. Page

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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