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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
61
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
71
Abbreviation of operation
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Alternation of maxima and minima
73
Modified method of finding maxima and minima
74
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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81
100
102
103
104
107
108
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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146
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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155
Examples of the application of Stirling's theorem
Extension of Taylor's theorem to functions of two variables
SECOND PART.
CHAPTER I.
Tangency.
159
99 Definition of a tangent and of a normal
160
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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168
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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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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
118
tangents at a multiple point
119 Multiplicity of a multiple point at the origin
Point of osculation
121
Remark on the theory of multiple points
122 Points d'arrêt or points de rupture
123 Points saillants
124
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
129
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
210
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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