A Synopsis of Elementary Results in Pure and Applied Mathematics

Volume 1
	PREFACE TO PART I
	TABLE OF CONTENTS. PART I
	INDEX TO PROPOSITIONS OF EUCLID
		Book I
		Book II, III
		Book IV, VI
		Book XI
	TABLE OF CONTENTS. PART II
	ERRATA. 1
	ERRATA. 2
	§1. MATHEMATICAL TABLES
		Introduction
		TABLE I-III
		TABLE IV-V
		TABLE VI-VIII
		Burckhakdt\'s Factor Tables
			1-9000
			9000-18000
			18000-27000
			27000-36000
			36000-45000
			45000-54000
			54000-63000
			63000-72000
			72000-81000
			81000-90000
			90000-99000
		Log Γ(n)
	§2. ALGEBRA
		Factors
			Newton\'s Rule for expanding a Binomial (12)
		Multiplication and Division, by the Method of Detached Coefficients
		Indices
		Highest Common Factor
		Lowest Common Multiple
		Evolution
		Quadratic Equations
		Theory of Quadratic Expressions
		Simultaneous Equations
		Ratio and Proportion
		Variation
		Arithmetical Progression
		Geometrical Progression
		Harmonical Progression
		Permutations and Combinations
		Surds
		Binomial Theorem
		Multinomial Theorem
		Logarithms
		Exponential Theorem
		Continued Fractions and Convergents
		Indeterminate Equations
		To reduce a Quadratic Surd to a Continued Fraction
			General Theory(199)
		Equations
			Miscellaneous Equations and Solutions (214)
		Imaginary Expressions
		Method of Indeterminate Coefficients
		Method of Proof by Induction
		Partial Fractions
		Convergency and Divergency of Series
		Expansion of a Fraction
		Recurring Series
		Summation of Series by the Method of Differences
		Direct Factorial Series
		Inverse Factorial Series
		Composite Factorial Series
		Miscellaneous Series
		Polygonal Numbers
		Figurate Numbers
		Hypergeometrical Series
		Interest
		Annuities
		Probabilities
		Inequalities
		Scales of Notation
		Theory of Numbers
	§3. THEORY OF EQUATIONS
		Factors of an Equation
		Descartes\' Rule of Signs
		The Derived Functions of f(x)
		Equal Roots of an Equation
		Limits of the Roots
		Newton\'s Method of Divisors
		Reciprocal Equations
		Binomial Equations
		Cubic Equations
		Biquadratic Equations
		Commensurable Roots
		Incommensurable Roots
		Symmetrical Functions of the Roots of an Equation
		Expansion of an Implicit Function of x
		Determinants
			General Theory (556)
			Analysis of a Determinant (568)
			Synthesis of a Determinant (569)
			Product of Two Determinants of the n-th Order (570)
			Partial and Complementary Determinants (576)
		Elimination
			I. Bezout\'s Method (586)
			II. Sylvester\'s Dialytic Method (587)
			III. Method of Elimination by Symmetrical Functions (588)
	§4. PLANE TRIGONOMETRY
		Angular Measurement
		Trigonometrical Ratios
		Formulae involving Two Angles, and Multiple Angles
		Ratios of Certain Angles
		Properties of the Triangle
		Solution of Triangles
		Regular Polygon and Circle
		Use of Subsidiary Angles
		de Moivre\'s Theorem
		Additional Fobmulae
			Examples of the Solution of Triangles (859)
	§5. SPHERICAL TRIGONOMETRY
		Introductory Theorems
		Right-angled Triangles
		Oblique-angled Triangles
		Spherical Triangle and Circle
		Spherical Areas
		Polyhedrons
	§6. ELEMENTARY GEOMETRY
		Miscellaneous Propositions
		The Nine-Point Circle
		Collinear and Concurrent Systems of Points and Lines
		Triangles of Constant Species Circumscribed to a Triangle
		Radical Axis
		The Method of Inversion
		Pole and Polar
		Coaxal Circles
		Centres and axes of similitude
		Anharmonic Ratio
		Homographic Systems of Points
		Involution
		The Method of Projection
		On Perspective Drawing
		Orthogonal Projection
		Projections of the Sphere
		Additional Theorems
	§7. GEOMETRICAL CONICS
		The Sections of the Cone
		The Ellipse and Hyperbola
		The Parabola
Volume 2
	TABLE OF CONTENTS. PART II.
	PREFACE TO PART II.
	§8. DIFFERENTIAL CALCULUS
		Introduction
		Differentiation
		Successive Differentiation
		Partial Differentiation
		Theory of Operations
		Expansion of Explicit Functions
		Expansion of Implicit Functions
		Indeterminate Forms
		Jacobians
		Quantics
		Implicit Functions
		Change of the Independent Variable
		Maxima and Minima
			Maxima and Minima values of a function of threeor more variables (1852)
			Continuous Maxima and Minima (1866)
	§9. INTEGRAL CALCULUS
		Introduction
		Methods of Integration
		Standard Integrals
		Various Indefinite Integrals
		Integration by Rationalization
		Integrals Reducible to Elliptic Integrals
		Successive Integration
		Hyperbolic Functions
		Definite Integrals
		Theorems Respecting the Limits of Integration
		Methods of Evaluating Definite Integeals
		Differentiation under the Sign of Integration
		Approximate Integration
		The Integrals B(l,m) and Γ(n)
		Integration of Algebraic Forms
		Integration of Logarithmic and Exponential Forms
		Integration of Circular Forms
		Integration of Circular Logarithmic and Exponential Forms
		Miscellaneous Theorems
			Frullani\'s Formula (2700)
			Poisson\'s Formulae (2702)
			Abel\'s Formula (2705)
			Kummer\'s Formula (2706)
			Cauchy\'s Formula (2712)
		Finite Variation of a Parameter
		Fourier\'s Formula
		The Function ψ(x)
		Numerical Calculation of log Γ(x)
		Change of the Variables in a Definite Multiple Integral
		Multiple Integrals
		Expansions of Functions in Converging Series
			Lagrauge\'s Method (2857)
			Miscellaneous Expansions (2911)
			Legendre\'s Function X_n (2936)
		Formulae for the Expansion of Functions in Trigonometrical Series
		Approximate Integeation
			Cotes\'s Method (2995)
			Gauss\'s Method (2997)
	§10. CALCULUS OF VARIATIONS
		Functions of One Independent Variable
			Particular Cases (3033)
			Other Exceptional Cases (3045)
		Functions of Two Dependent Vaeiables
			Relative Maxima and Minima (3069)
			Geometrical Applications (3070)
		Functions of Two Independent Variables
			Geometrical Applications (3078)
		Appendix
			On the General Object of the Calculus of Variations (3084)
			Successive Variation (3087)
			Immediate Intesuability of the Function V (3090)
	§11. DIFFERENTIAL EQUATIONS
		Generation of Differential Equations
		Definitions and Rules
		Singular Solutions
		Geometrical Meaning of a Singular Solution (3072)
		Determination of a Singular Solution from the Differential Equation (3075)
		Rules for Discriminating a Singular Solution of the Envelope Species (3078)
		First Order Linear Equations
		First Order Non-linear Equations
		Higher Order Linear Equations
		Higher Order Non-linear Equations
		Exact Differential Equations
		Miscellaneous Methods
		Singular Solutions of Higher Order Equations
		Equations with more than Two Variables
		Simultaneous Equations with One Independent Variable
		Partial Differential Equations
		Second Order P. D. Equations
			Poisson\'s Equation (3441)
			Laplace\'s Reduction of the Equation (3442)
		Law of Reciprocity [Boole, ch. xv
		Symbolic Methods
		Solution of Linear Differential Equations by Series
		Solution by Definite Integrals*
		P. D. Equations with more than Two Independent Variables
		Differential Resolvents of Algebraic Equations
	§12. CALCULUS OF FINITE DIFFERENCES
		Introduction
		Formulae for First and n-th Differences
		Interpolation
		Mechanical Quadrature
		Summation of Series
		Appeoximate Summation
	§13. PLANE COORDINATE GEOMETRY
		Systems of Coordinates
		ANALYTICAL CONICS IN CARTESIAN COORDINATES
			Lengths and Areas
			Transformation of Coordinates
			The Right Line
			General Methods
			The Circle
			The Parabola
			The Ellipse and Hyperbola (See also p.233, et seq.)
			The Hyperbola referred to the Asymptotes
			The General Equation
			Similar Conics
			Circle of Curvature
			Confocal Conics
		ANALYTICAL CONICS IN TRILINEAR COORDINATES
			The Right Line
			Anharmonic Ratio
			The General Equation of a Conic
			Particular Conics
				The Circumscribing Conic of the Trigon (4724)
				The Circumscribing Circle of the Trigon (4738)
				The Inscribed Conic of the Trigon (4739)
				The Inscribed Circle of the Trigon (4747)
				General Equation of the Circle (4751)
				The Nine-point Circle (4754)
				The Triplicate-ratio Circle (4754b)
				The Seven-point Circle* (4754c)
			Conic and Self-conjugate Triangle
			Important Theorems
				Carnot\'s Theorem (4778)
				Pascal\'s Theorem (4781)
				Brianchon\'s Theorem (4783)
			The Conic Referred to Two Tangents and the Chord of Contact
			Anharmonic Pencils of Conics
			Construction of Conics
			The Method of Reciprocal Polars
				General Rules for Reciprocating (4846)
			Tangential Coordinates
			On the Intersection of Two Conics
			The Method of Projection
			Invariants and Covariants
				To Find the Foci of the General Conic (4656)
				Note on Tangential Coordinates (5030)
		THEORY OF PLANE CURVES
			Tangent and Normal
			Radius of Curvature and Evolute
			Inveese Problem and Intrinsic Equation
			Asymptotes
			Singularities of Curves
			Contact of Curves
			Envelopes
			Integrals of Curves and Areas
			Inverse Curves
			Pedal Curves
			Roulettes
			Transcendental and other Curves
			Linkages and Linkwork
			Mechanical Calculators
			Appendix on Biangular Coordinates*
	§14. SOLID COORDINATE GEOMETRY
		Systems of Coordinates
		Transformation of Coordinates
		The Sphere
		Cylindrical and Conical Surfaces
		Conicoids
		Central Quadric Surface
		The General Equation of a Quadric
		Reciprocal Polars
			Rules for Reciprocating (5705)
		Theory of Tortuous Curves
		General Theory of Surfaces
		Invariants
		Integrals for Volumes and Surfaces
		Centre of Mass
		Moments and Products of Inertia
		Perimeters, Areas, Volumes, Centres of Mass, and Moments of Inertia of Various Figures
	JOINT INDEX
		Key to the Index
		Explanation of Abbreviations, &c.
		Index
			A
			B
			C
			D
			E
			F
			G
			H
			I
			JK
			L
			M
			N
			O
			P
			Q
			R
			S
			T
			UV
			WZ
	Fig. 1-15
	Fig. 16-30
	Fig. 31-45
	Fig. 46-51
	Fig. 52-62
	Fig. 63-75
	Fig. 76-86
	Fig. 87-88
	Fig. 89-93
	Fig. 94-110
	Fig. 111-115
	Fig. 116-123
	Fig. 124-130
	Fig. 133-141
	Fig. 142-152
	Fig. 153-164
	Fig. 165-167
	Fig. 168-177
	Fig. 178-182
	Fig. 183-193