103

Enumeration of Methods, .

92

112

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

94-99

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

125

126-128 Failure of Taylor's and Maclaurin's Theorems,

129

121-122 Formulae of Cauchy and Schlömilch and Roche,
123-124 Application to Maclaurin's Theorem and Special

Cases of Taylor's Theorem,

Geometrical Illustration of Lagrange-Formula,

Examples of Application of Lagrange-Formula,

109-110

110

110-111

111-114

114-115

ARTS.

145

Differentiation of an Implicit Function,

146-150 Order of Partial Differentiations Commutative,

PAGES.

135

135-138

151-152 Second Differential Coefficient of an Implicit Function, 138-139

An Illustrative Process,

153
154-155 An Important Theorem,

141

141-143

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

143-145

145-152

Cartesians and Polars,

174-178 Geometrical Results.

179-181 Polar Subtangent, Subnormal, etc.,

182-183 Polar Equations of Tangent and Normal,

184-186 Number of Tangents and Normals from a given

159-161

161-163

164-165

165-169

169-171

171-172

172-174

174-175

175-177

177-181

181

181-186

195-199 Important Geometrical Results,

200 Tangential Equation,

186-187

201-204 Inversion,

205-207 Polar Reciprocals,

CHAPTER VIII.

ASYMPTOTES.

208-210 To find the Oblique Asymptotes,

211-213 Number of Asymptotes to a Curve of the nth degree,

214

215

216

Asymptotes parallel to the Co-ordinate Axes,
Method of Partial Fractions for Asymptotes,
Particular Cases of the General Theorem,
217-218 Limiting Form of Curve at Infinity,
219-220 Asymptotes by Inspection,

226-229 Curve in general on opposite sides of the Asymptote

230

at opposite extremities. Exceptions,

Curvilinear Asymptotes,

231-233 Linear Asymptote obtained by Expansion,

234-235 Polar Equation to Asymptote,

PAGES.

214

215

215-216

216-218

219

219-221

221-224

224

236

Circular Asymptotes,

254-257 To examine the Nature of a specified point on a Curve, 242-248

264

258-259 To discriminate the Species of a Cusp,
260-261 Singularities of Transcendental Curves,
262-263 Maclaurin's Theorem with regard to Cubics,
Points of Inflexion on a Cubic are Collinear,

CHAPTER X.

CURVATURE.

265-266 Angle of Contingence. Average Curvature,

265-266

267-268 Curvature of a Circle. Radius of Curvature,

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,

304

305

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

General Investigation of Equation to Envelope,

306-307 Envelope of AX2+2Bλ + C = 0, .

.308-311

312

311

311-312

312-313

313-314

315-318

Several Parameters. Indeterminate Multipliers,
Converse Problem. Given the Family and the Enve-
lope to find the Relation between the Parameters, 318-320

315-317 Nature of the Problem; Order of Procedure in

326

323-325 Curves of the Classes r = a sin no, r sin no
Curves of the Class = a" cos no. Spirals,

APPLICATION TO THE EVALUATION OF

SINGULAR FORMS AND MAXIMA

AND MINIMA VALUES.