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CONTENTS.

PRINCIPLES AND PROCESSES OF THE

DIFFERENTIAL CALCULUS. -

CHAPTER I.

DEFINITIONS. LIMITS.
ARTS.
1-2 Object of the Calculus,
3-9 Definitions,
10-12 Limits. Illustrations and Fundamental Principles,
13-17 Undetermined Forms,
18-24 Four Important Undetermined Forms,
25 Hyperbolic Functions,
26-36 Infinitesimals, .

PAGES.

1
2-5
5-9
9.11
12-15
16-17
17-24

CHAPTER II.

FUNDAMENTAL PROPOSITIONS.
37-38 Tangent to a Curve,
39-41 Differential Coefficients, Examples, Notation, .
42-44 Aspect as a Rate-Measurer,
45-54 Constant, Sum, Product, Quotient, .
55-57 Function of a Function,
58-62 Inverse Functions,

26-30
30-34
34-36
37-41
42-44
44-48

CHAPTER III.

STANDARD FORMS.
63-67 Differentiation of x”, ar, log x,
68-73 The Circular Functions, .
74-81 The Inverse Trigonometrical Functions,
82 Interrogative Character of the Integral Calculus,

49-51
51-53
54-57

58

ARTS.
83

84
85
86-87

Table of Results to be remembered, .
Cases of the Form u',
Hyperbolic Functions. Results,
Illustrations of Differentiation,

PAGES.
59-60
60-61
61-62
62-65

72-73

CHAPTER IV.

SUCCESSIVE DIFFERENTIATION.
88-89 Repeated Differentiations,

d
90-94 as an Operative Symbol,


95-97 Standard Results and Processes,
98-100 Leibnitz's Theorem,
101 Note on Partial Fractions,

74-78

78-82
82-85
85-88

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CHAPTER V.

EXPANSIONS.
102 Enumeration of Methods, .

92
103 Method I.-Algebraical and Trigonometrical Methods, 92-94
104-111 Method II.-Taylor's and Maclaurin's Theorems, 94-99
112 Method III.—Differentiation or Integration of known
Series,

99-101
113 Method IV.-By a Differential Equation,

101-103
114-119 Continuity and Discontinuity, ·

104-107
120 Lagrange-Formula for Remainder after n Terms of
Taylor's Series,

107-109
121-122 Formulae of Cauchy and Schlömilch and Roche, 109-110
123-124 Application to Maclaurin's Theorern and Special
Cases of Taylor's Theorem,

110
125 Geometrical Illustration of Lagrange-Formula, 110-111
126-128 Failure of Taylor's and Maclaurin's Theorems, 111-114
129 Examples of Application of Lagrange-Formula, 114-115
130 Bernoulli's Numbers,

115-117

CHAPTER VI.

PARTIAL DIFFERENTIATION.
132-134 Meaning of Partial Differentiation,
135-136 Geometrical Illustrations,
137-139 Differentials,
140-144 Total Differential and Total Differential Coefficient,

126-127
127-129
129-132
132-134

PAGES.

135

ARTS.
145 Differentiation of an Implicit Function,
146-150 Order of Partial Differentiations Commutative, 135-138
151-152 Second Differential Coefficient of an Implicit Function, 138-139
153 An Illustrative Process,

141
154-155 An Important Theorem,

141-143
156-160 Extensions of Taylor's and Maclaurin's Theorems, 143-145
161-168 Homogeneous Functions. Euler's Theorems,

145-152

APPLICATIONS TO PLANE CURVES.

159-161
161-163
164-165
165-169
169-171
171-172

CHAPTER VII.

TANGENTS AND NORMALS.
169-171 Equation of Tangent in various Forms,
172 Equation of Normal,
173 Tangents at the Origin,
174-178 Geometrical Results. Cartesians and Polars,
179-181 Polar Subtangent, Subnormal, etc., .
182-183 Polar Equations of Tangent and Normal,
184-186 Number of Tangents and Normals from a given

point to a Curve of the nth degree,
187 Polar Line, Conic, Cubic, etc., .
188-190 Pedal Equation of a Curve,
191-193 Pedal Curves,
194 Tangential-Polar Equation,
195-199 Important Geometrical Results,
200 Tangential Equation,
201-204 Inversion,
205-207 Polar Reciprocals,

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172-174
174-175
175-177
177-181

181
181-186
186-187
187-190
190-192

CHAPTER VIII.

ASYMPTOTES.
208-210 To find the Oblique Asymptotes,
211-213 Number of Asymptotes to a Curve of the nth degree,
214 Asymptotes parallel to the Co-ordinate Axes,
215 Method of Partial Fractions for Asymptotes,
216 Particular Cases of the General Theorem,
217-218 Limiting Form of Curve at Infinity,
219-220 Asymptotes by Inspection,

203-205

206
206-208
208-209
209-211
211-213
213-214

.

PAGES.

ARTS.
21

214

215
215-216

Curve through points of intersection of a given curve

with its Asymptotes,
222

Newton's Theorem,
223-225 Other Definitions of Asymptotes,
226-229 Curve in general on opposite sides of the Asymptote

at opposite extremities. Exceptions,
230 Curvilinear Asymptotes,
231-233 Linear Asymptote obtained by Expansion,
234-235 Polar Equation to Asymptote, ·
236 Circular Asymptotes,

216-218

219
219-221
221-224

224

CHAPTER IX.

SINGULAR POINTS.

237 Concavity and Convexity,

229
238-240 Points of Inflexion and Undulation, .

229-231
241-248 Analytical Conditions,

231-238
249-250 Multiple Points,

238-240
251-253 Double Points,

240-242
254-257 To examine the Nature of a specified point on a Curve, 242-248
258-259 To discriminate the Species of a Cusp,

248-253
260-261 Singularities of Transcendental Curves,

254-256
262-263 Maclaurin's Theorem with regard to Cubics,

256-257
264 Points of Inflexion on a Cubic are Collinear,

257-258

CHAPTER X.

CURVATURE.

265-266 Angle of Contingence. Average Curvature,
267-268 Curvature of a Circle. Radius of Curvature,
269-271 Formula for Intrinsic Equations,
272-275 Formulae for Cartesian Equations,
276-279 Curvature at the Origin,
280 Formula for Pedal Equations,
281-282 Formulae for Polar Curves,
283 Tangential-Polar Formula,
284 Conditions for a Point of Inflexion,
285

Co-ordinates of Centre of Curvature,
286-290 Involutes and Evolutes,
291-294 Intrinsic Equations, ·

265-266
266-268
268-269
269-272
273-275
276-277
277-278

278
278-279
279-281
281-285
285-288

ARTS.

PAGES.
295-297 Contact. Analytical Conditions,

288-293
298 Osculating Circle,

293-294
299-300 Conic of Closest Contact, .

294-297
301 Tangent and Normal as Axes; x and y in terms of s, 297-298

CHAPTER XI.

ENVELOPES.
302-303 Families of Curves ; Parameter ; Envelope,

311
304 The Envelope touches each of the Intersecting Mem-
bers of the Family, .

311-312
305 General Investigation of Equation to Envelope, 312-313
306-307 Envelope of A12+2BX+C =0, .

313-314
308-311 Several Parameters. Indeterminate Multipliers, 315-318
312 Converse Problem. Given the Family and the Enve-

lope find the Relation between the Parameters, 318-320
313 Evolutes, .

320
314 Pedal Curves,

321-322
CHAPTER XII.

CURVE TRACING.
315-317 Nature of the Problem ; Order of Procedure in
Cartesians,

330-333
318-319 Examples,

333-340
320-321 Newton's Parallelogram,

340-344
322 Order of Procedure for Polar Curves,

344-345
323-325 Curves of the Classes r= a sin no, » sin no= a,

345-347
326 Curves of the Class yn = a" cos no. Spirals,

347-352

APPLICATION TO THE EVALUATION OF
SINGULAR FORMS AND MAXIMA

AND MINIMA VALUES.

CHAPTER XIII.

UNDETERMINED Forms.
327-329 Forms to be discussed,
330 Algebraical Treatment,
331-334 Form

o'
335

Form 0.xco,

361-362
362-365

365-369

369

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