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Partial differentiation of an explicit function of three variables,
one of which is a function of the other two
Partial differentiation of an explicit function of n +r variables,
r independent and n dependent
27 Partial differentiation of an implicit function of two independent
variables
28 Partial differentiation of implicit functions of any number of
independent variables
43 Differential coefficient of sec-1x
44 Differential coefficient of cosec-1x
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Differentiation of simple functions of y with regard to x
45' Illustrative examples
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Successive differentiation of an explicit function of two func-
tions of a single variable
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50 Successive differentiation of an implicit function of a single
Successive differentiation of an explicit function of three vari-
ables, one of which is a function of the other two
Change of variables
Transformation of one system of independent variables into
another
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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
Elimination of arbitrary functions when the number of indepen-
dent variables exceeds two
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CHAPTER V.
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Evaluation of Indeterminate Functions.
63 Indeterminateness of explicit functions of a single variable
Evaluation of functions of the form
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Failure of the method of differentials for the evaluation of in-
determinate functions
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Evaluation of indeterminate functions of several independent
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Evaluation of indeterminate implicit functions of a single variable 97
75 Maxima and, minima of implicit functions of a single variable
76 Maxima and minina of a function of a function
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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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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CHAPTER VII.
Development of Functions.
86 Examples of Taylor's theorem
87,88 Failure of Taylor's theorem .
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Lagrange's theory of Functions
Stirling's theorem .
91 Examples of the application of Stirling's theorem
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92 Extension of Taylor's theorem to functions of two variables 93 Failure of the development of f(x+h, y+k) by Taylor's theorem Limits and remainders of the development of ƒ (x + h, y + k)
95 Example of the application of Taylor's theorem for two variables 150
96 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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SECOND PART.
CHAPTER I.
Tangency.
Inclinations of the tangent and the normal at any point of a
curve to the coordinate axes
160
101 Equations to the tangent and the normal at any point of a
102 Distance of the origin of coordinates from the tangent
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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114
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
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117 Analytical property of conjugate points in algebraical curves
118 Determination of the multiplicity and of the directions of the
tangents at a multiple point
119 Multiplicity of a multiple point at the origin
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
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On the Index of Curvature, the Radius of Curvature, and the Centre
of Curvature, of a Plane Curve.
Radius and centre of curvature
130 Expression for p when x is the independent variable
131 Expressions for p when s is the independent variable
132 Expression for p in terms of dx, dy, d'x, d'y
133 Expression for p in terms of partial differential coefficients
134 Another method of finding the radius of curvature
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ρ
CHAPTER VI.
Analytical Determination of the Centre of Curvature. Theory of Evolutes
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
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terms of partial differential coefficients of u
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137 Locus of the centre of curvature
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The normal at any point of the involute a tangent at the
corresponding point of the evolute
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139 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
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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
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146 General case of any number of parameters
Intersection of consecutive normals to a curve
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CHAPTER IX.
Differentials of Areas, Volumes, Arcs, and Surfaces.
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