103 Enumeration of Methods, . 92 112 Method I.-Algebraical and Trigonometrical Methods, 92-94 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 107-109 125 126-128 Failure of Taylor's and Maclaurin's Theorems, 129 121-122 Formulae of Cauchy and Schlömilch and Roche, 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, 153 141 141-143 156-160 Extensions of Taylor's and Maclaurin'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, 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, 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- 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, 315-317 Nature of the Problem; Order of Procedure in 326 323-325 Curves of the Classes r = a sin no, r sin no APPLICATION TO THE EVALUATION OF SINGULAR FORMS AND MAXIMA AND MINIMA VALUES. |