About this book
57 Elimination of irrational, logarithmic, exponential, and circular
functions of known functions
69
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
72
74
77
81
63
64
CHAPTER V.
Evaluation of Indeterminate Functions.
Indeterminateness of explicit functions of a single variable
Evaluation of functions of the form
65 Failure of the method of differentials for the evaluatiou of in
determinate functions
84
89
92
66 Evaluation of indeterminate functions of several independent
variables
93
67 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
119
81 Application of indeterminate multipliers to problems of maxima
and minima
122
Article
82, 83 Taylor's theorem
CHAPTER VII.
Development of Functions.
84 Another demonstration of Taylor's theorem
85 Cauchy's expression for R,
86 Examples of Taylor's theorem
87,88 Failure of Taylor's theorem .
89 Lagrange's theory of Functions
90 Stirling's theorem .
91
Page
128
133
134
136
137
140
141
146
149
Examples of the application of Stirling's theorem
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
94

151
97 Lagrange's formula for the development of implicit functions
98 Laplace's formula for the development of implicit functions
Inclinations of the tangent and the normal at any point of a
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 a and y
108 Oblique axes
166
168
CHAPTER II.
Asymptotes.
109 Definition of an asymptote. Method of finding asymptotes
110,111 Asymptotes of algebraic curves
170
171
Multiple Points, Conjugate Points, Cusps, &c.
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
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
Multiplicity of a multiple point at the origin
120
Point of osculation
121
Remark on the theory of multiple points
122 Points d'arrêt or points de rupture
123 Points saillants
Concavity and Convexity of Curves and Points of Inflection.
125 Conditions for concavity and convexity
126 Condition for a point of inflection
127
178
179
181
182
185
186
187
189
191
192
194
197
On the Index of Curvature, the Radius of Curvature, and the Centre
of Curvature, of a Plane Curve.
129 Radius and centre of curvature
Expression for p when x is the independent variable
203
130
204
131 Expressions for when s is the independent variable
132 Expression for p in terms of dx, dy, d2x, d'y
206
207
133 Expression for p in terms of partial differential coefficients
134 Another method of finding the radius of curvature
209
P
CHAPTER VI.
Analytical Determination of the Centre of Curvature. Theory of Evolutes
and Involutes.
135
Determination of the coordinates of the centre of curvature
Formulæ for the coordinates of the centre of curvature in
210
terms of partial differential coefficients of u
211
137 Locus of the centre of curvature
212
138
The normal at any point of the involute a tangent at the
corresponding point of the evolute
213
139
Generation of the involute by the end of a thread unwound
from the evolute
214
To find the length of any arc of the evolute of a curve
215
Contact of Curves.
Definition of order of contact
217
142
143
144
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,
218
the contact is of the third order
221
CHAPTER VIII.
Envelops.
Differentials of Areas, Volumes, Arcs, and Surfaces.
161 Conditions for the concavity and convexity of the curve
towards the pole and for points of inflection.
CHAPTER XI.
On the Methods of tracing the forms of Curves from their Equations.