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54-66. The roots of + 1 and of — 1 108

67. Imaginary logarithms 113

Section 3.—Theory of the equicrescent variable, and Taylor's Theorem.

68. The relation between y and its equivalent/(x), of the equation

y = f (x). when x has been successively increased n

times 115

69. 70. Modification of the preceding, when * is equicrescent .. 116

71. Taylor's Series 119

72. Limits of Taylor's Series 119

73. Examples of Taylor's Series 121

74. Expansion of /(*) in ascending powers of x—a 123

Section 4.—Change of equicrescent variable, and transformation of

differential expressions.

75. Different forms of the problem 123

76. Requisite formulae for a function of two variables 124

77- Transformations in terms of a new variable 125

Section 5.—Successive differentiation of functions of many

independent variables.

78. Explanation of the symbols 128

79. The order of successive differentiations with respect to many

variables is indifferent 131

80. 81. Application of the principles of the preceding Articles to

functions of two and more variables 133

82. Euler's Theorems of homogeneous functions 137

83. Transformation of partial derived functions 139

Section 6.—Successive differentiation of implicit functions.

84. Calculation of derived-functions when the relation between

the variables is given in an implicit form .. 141

85. Examples of preceding formula? 142

86. Expansion of one of the variables of an implicit function in

terms of the other by means of Maclaurin's Theorem . . 143

87- Calculation and properties of Bernoulli's numbers 145

88. Examples of Bernoulli's numbers 146

89. The sums of the powers of the reciprocals of the natural

numbers 148

90. Lagrange's Theorem 149

91. Examples of Lagrange's Theorem 153
92. Expansion of f(x) in ascending powers of <f>(x) 135
114. If in the theorem of the last Article J\x) = (x— x,i)", then

Section 2.—Functions of many variables.

137- The theorem for functions of two variables analogous to

that of Art. 111 222

138. A particular form of the preceding 224

139. Evaluation of indeterminate quantities which are functions

of two independent variables 225

140. Expansion of F(x + h, y + k) 228

141. Examples of the preceding 229

142. Expansion of F (x+h, y + A, z +I ) 230

143. Extension of the process of derivation to functions of two

and more variables 233

CHAPTER VII.

MAXIMA AND MINIMA.

144. Definition of maximum and minimum 235

Section 1.—Maxima and minima of explicit functions of one variable.

145. Method of determining such singular values of f(x) 236

146. Easy detection of the singular values when f(x) is alge-

braical 237

147- Geometrical representation of the criteria 238

148. Examples illustrative of the preceding methods 238

149, 150. The forms of the derived-functions of f(x), when it

has such singular values; and a method of thereby deter-

mining maxima and minima of f(x) 240

151. Examples of maxima and minima 243

152. The determination of the number of maxima and minima of

a given function 247

153. The absolute maximum and minimum 249

Section 2.—Maxima and minima of implicit functions of two

independent variables.

154. Method of determining such values, and examples 250

Section 3.—Maxima and minima of an explicit function of two

indejtendent variables.

155. Definition of maxima and minima of functions of two vari-

ables 252

156. 157- Method of determining criteria of such singular values 253

158. Another method of determining such singular values .... 255

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