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CHAPTER IV.

Elimination of Constants and Functions.

Elimination of constants

Partial elimination of constants

Elimination of irrational, logarithmic, exponential, and circular

functions of known functions

58, 59 Elimination of an arbitrary function of a known function

60 Elimination of any number of arbitrary functions of known

functions

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Elimination of arbitrary functions of unknown functions

62 Elimination of arbitrary functions when the number of indepen-

dent variables exceeds two

CHAPTER V.

CHAPTER VI.

Maxima and Minima.

Definition of a maximum and minimum

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69 Lemma

70 Rule for finding maxima and minima

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Abbreviation of operation

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Alternation of maxima and minima

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Modified method of finding maxima and minima

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75 Maxima and, minima of implicit functions of a single variable

76 Maxima and minina of a function of a function

Evaluation of Indeterminate Functions.

63 Indeterminateness of explicit functions of a single variable

64 Evaluation of functions of the form

65 Failure of the method of differentials for the evaluatiou of in- determinate functions

66 Evaluation of indeterminate functions of several independent

variables

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67 Evaluation of indeterminate implicit functions of a single variable 97

80 Maxima and minima corresponding to indeterminate differential

coefficients

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81 Application of indeterminate multipliers to problems of maxima

and minima

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100

102

103

104

107

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112

77, 78 Maxima and minima of a function of two independent variables 114

79 Maxima and minima of functions of any number of independent

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85 Cauchy's expression for R, .

86 Examples of Taylor's theorem

87,88 Failure of Taylor's theorem .

Lagrange's theory of Functions

Stirling's theorem .

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Failure of the development of f(x+h, y+k) by Taylor's theorem 149

Limits and remainders of the development of ƒ (x + h, y + k)

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95 Example of the application of Taylor's theorem for two variables 150

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151

Stirling's theorem applied to functions of two variables

97 Lagrange's formula for the development of implicit functions

Laplace's formula for the development of implicit functions

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Examples of the application of Stirling's theorem

Extension of Taylor's theorem to functions of two variables

SECOND PART.

CHAPTER I.

Tangency.

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99 Definition of a tangent and of a normal

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100 Inclinations of the tangent and the normal at any point of a

curve to the coordinate axes

101 Equations to the tangent and the normal at any point of a

curve

102 Distance of the origin of coordinates from the tangent

103 Intercepts of the tangent

104 Subtangent

105 Length of the tangent

106 Normal and subnormal

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107 Form of the equation to the tangent to curves of which the

equations involve only rational functions of x and y

108 Oblique axes

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CHAPTER II.

Definition of an asymptote. Method of finding asymptotes 170

110,111 Asymptotes of algebraic curves

Examples of asymptotes

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173

113 Algebraical method of finding curvilinear and rectilinear

asymptotes

Branches pointillées

Asymptotes.

CHAPTER III.

114

Multiple Points, Conjugate Points, Cusps, &c.

Definition of multiple points, conjugate points, and cusps

115 Analytical property of multiple points in algebraical curves

116 Analytical property of cusps in algebraical curves

117 Analytical property of conjugate points in algebraical curves

Determination of the multiplicity and of the directions of the

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tangents at a multiple point

119 Multiplicity of a multiple point at the origin

Point of osculation

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Remark on the theory of multiple points

122 Points d'arrêt or points de rupture

123 Points saillants

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Concavity and Convexity of Curves and Points of Inflection.

125 Conditions for concavity and convexity

126 Condition for a point of inflection

127 Symmetrical investigation of points of inflection

128 Index of curvature

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Radius and centre of curvature

variable

130 Expression for p when x is the independent variable

131 Expressions for when s is the independe

132 Expression for p in terms of dx, dy, d3x, d3y

133 Expression for p in terms of partial differential coefficients

134 Another method of finding the radius of curvature

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On the Index of Curvature, the Radius of Curvature, and the Centre

of Curvature, of a Plane Curve.

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Analytical Determination of the Centre of Curvature.

and Involutes.

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Determination of the coordinates of the centre of curvature

Formulæ for the coordinates of the centre of curvature in

terms of partial differential coefficients of u

Locus of the centre of curvature

The normal at any point of the involute a tangent at the

corresponding point of the evolute

Generation of the involute by the end of a thread unwound

from the evolute

To find the length of any arc of the evolute of a curve

CHAPTER VII.

Contact of Curves.

CHAPTER VIII.

Definition of order of contact

The higher the order of contact, the closer the contact

Order of contact dependent upon the number of parameters

When the radius of curvature is a maximum or a minimum,

the contact is of the third order

Envelops.

Case of a single parameter

General case of any number of parameters

Intersection of consecutive normals to a curve

CHAPTER IX.

Theory of Evolutes

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148 Differential of an area

149 Differential of a volume of revolution

150 Differential of an arc

151 Differential of a surface of revolution

Differentials of Areas, Volumes, Arcs, and Surfaces.

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152 Tangency

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CHAPTER X.

Curves referred to Polar Coordinates,

Differential of an area

154 Diagram of differentials

155 Radius of curvature in terms of r and p

Chord of curvature through the pole

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157 Radius of curvature in terms of r and

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Evolutes of polar curves

Asymptotes

160 Asymptotic circles

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Conditions for the concavity and convexity of the curve

towards the pole and for points of inflection.

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CHAPTER XI.

On the Methods of tracing the forms of Curves from their Equations.

162 General principles and examples

163 Homogeneous curves

164 The cycloid

165 Tangent and normal to the cycloid

Arc of the cycloid

167

Radius of curvature of the cycloid.

168 Evolute of the cycloid.

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