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Partial differentiation of an explicit function of three variables,
one of which is a function of the other two
21
26 Partial differentiation of an explicit function of n + r variables,
r independent and n dependent
22
27 Partial differentiation of an implicit function of two independent
variables
23
28 Partial differentiation of implicit functions of any number of
independent variables
29 Simple functions
30 Differential coefficient of ac"
31 Differential coefficient of loga X
26
32 Differential coefficient of qx
28
33 Differential coefficient of sin a
34 Differential coefficient of cos x
35 Differential coefficient of tan x
30
36 Differential coefficient of cot x
37 Differential coefficient of sec x
31
38 Differential coefficient of cosec x
39 Differential coefficient of sin ??
32
40 Differential coefficient of cos'x
33
41 Differential coefficient of tan'x
42 Differential coefficient of cot'x
34
43 Differential coefficient of sec'x
44 Differential coefficient of cosec'x
35
45 Differentiation of simple functions of y with regard to x
45' Illustrative examples
37
CHAPTER III.
44
46
50
52
Successive Differentiation.
46 Theory of the independent variable
47 Change of the independent variable
48 Order of partial differentiations indifferent
49 Successive differentiation of an explicit function of two func
tions of a single variable
50 Successive differentiation of an implicit function of a single
variable
51 Successive total differentials
52 Successive differentiation of an explicit function of three vari
ables, one of which is a function of the other two
53 Change of variables
54 Transformation of one system of independent variables into
another
55
57
а
58
60
61
CHAPTER IV.
67
68
Elimination of Constants and Functions.
55 Elimination of constants
56 Partial elimination of constants
57 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
69
72
74
77
81
CHAPTER V.
Evaluation of Indeterminate Functions.
63 Indeterminateness of explicit functions of a single variable 84
64 Evaluation of functions of the form o
89
65 Failure of the method of differentials for the evaluatiou of in
determinate functions
92
66 Evaluation of indeterminate functions of several independent
93
67 Evaluation of indeterminate implicit functions of a single variable 97
CHAPTER VI.
Maxima and Minima.
68 Definition of a maximum and minimum
99
69 Lemma
70 Rule for finding maxima and minima
100
71 Abbreviation of operation
102
72 Alternation of maxima and minima
103
73 Modified method of finding maxima and minima
104
74 Abbreviation of operation
107
75 Maxima and, minima of implicit functions of a single variable 108
76 Maxima and minina of a function of a function
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
119
80 Maxima and minima corresponding to indeterminate differential
coefficients
120
81 Application of indeterminate multipliers to problems of maxima
and minima
122
a
CHAPTER VII.
Development of Functions.
82, 83 Taylor's theorem
128
84 Another demonstration of Taylor's theorem
133
85 Cauchy's expression for R,
134
86 Examples of Taylor's theorem
136
87, 88 Failure of Taylor's theorem .
89 Lagrange's theory of Functions
137
90 Stirling's theorem .
140
91 Examples of the application of Stirling's theorem
141
92 Extension of Taylor's theorem to functions of two variables 146
93 Failure of the development of f(x+h, y+k) by Taylor's theorem 149
94 Limits and remainders of the development of f (x + h, y + k)
95 Example of the application of Taylor's theorem for two variables 150
96 Stirling's theorein applied to functions of two variables
151
97 Lagrange's formula for the development of implicit functions 152
98 Laplace's formula for the development of implicit functions 155
SECOND PART.
CHAPTER I.
Tangency.
.
curve
99 Definition of a tangent and of a normal
159
100 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
162
102 Distance of the origin of coordinates from the tangent
164
103 Intercepts of the tangent
165
104 Subtangent
105 Length of the tangent
166
106 Normal and subnormal
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
168 CHAPTER II.
Asymptotes.
109 Definition of an asyınptote. Method of finding asymptotes 170
110,111 Asymptotes of algebraic curves
171
112 Examples of asymptotes
173
113 Algebraical method of finding curvilinear and rectilinear
asymptotes
175
178
179
181
Multiple Points, Conjugate Points, Cusps, 8c.
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
119 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
124 Branches pointillées
Concavity and Condexity 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
192
194
197
On the Index of Curvature, the Radius of Curvature, and the Centre
of Curvature, of a Plane Curve.
128 Index of curvature
203
129 Radius and centre of curvature
130 Expression for p when x is the independent variable
204
131 Expressions for p when s is the independent variable
206
132 Expression for p in terms of dx, dy, dox, d'y
207
133 Expression for p in terms of partial differential coefficients
134 Another method of finding the radius of curvature
209
р
S
Analytical Determination of the Centre of Curvature. Theory of Evolutes
and Involutes.
135 Determination of the coordinates of the centre of curvature 210
136 Formulæ for the coordinates of the centre of curvature in
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
140 To find the length of any arc of the evolute of a curve 215
Contact of Curves.
141 Definition of order of contact
217
142 The higher the order of contact, the closer the contact 218
143 Order of contact dependent upon the number of parameters
144 When the radius of curvature is a maximum or a minimum,
the contact is of the third order
221
CHAPTER VIII.
Envelops.
145 Case of a single parameter.
146 General case of any number of parameters
147 Intersection of consecutive normals to a curve
223
226
230
CHAPTER IX.
Differentials of Areas, Volumes, Arcs, and Surfaces.