VIII.

ARTICLE.

SYNOPSIS OF LAWS OF ALGEBRAIC OPERATION.

31. Law of Signs.

32-36. Laws of association, commutation, distribution, ex-

ponentiation and logarithmication

37. Properties of o, I and ∞o

38. Agenda: Involution and logarithmic operation in arith-

metic. .

CHAPTER II.

GONIOMETRIC AND HYPERBOLIC RATIOS.

IX.

GONIOMETRIC RATIOS.

39. Definition of arc-ratio . . .

40. Definitions of goniometric ratios.

41. Agenda: Properties of goniometric ratios

42. Line equivalents of goniometric ratios.

43. Proof that limit (sin 0) / (= 1, when ✪ = 0

44. Area of a circular sector

45. Agenda: The addition theorem for goniometric ratios.

46. Definitions of hyperbolic ratios

47. Agenda: Properties of hyperbolic ratios.

48. Geometrical construction for hyperbolic ratios

49. Agenda: Properties of the equilateral hyperbola

52. Proof that limit (sinh u) | u=1, when u ___

53. Area of a hyperbolic sector

55. Approximate value of natural base . .

54. Agenda: The addition theorem for hyperbolic ratios.

56. Agenda: Logarithmic forms of inverse hyperbolic ratios 65

59

60

62

63

CHAPTER III.

THE ALGEBRA OF COMPLEX QUANTITIES.

ARTICLE.

XI. GEOMETRIC ADDITION AND MULTIPLICATION.

57. Classification of magnitudes: Definitions.

58. Geometric addition .

PAGE.

59. Commutative and associative laws for geometric addition 69

60. Geometric multiplication

61. Conjugate and reciprocal

62. Agenda: Properties of cis .

63. The imaginary unit . .

64. Commutative and associative laws for geometric multipli-
cation. . . .

66

74

75

76

XII. EXPONENTIALS AND LOGARITHMS.

68. Definitions of modulus, base, exponential, logarithm

69. Exponential of o, I and logarithm of 1, B.

70. Classification of logarithmic systems.

71. Special constructions

72. Correspondence of initial values of exponential and

logarithm.

83

84

82. Many-valuedness of logarithms

83. Direct and inverse processes.

84. Agenda: Reduction of exponential and logarithmic forms 94

89

89

90

90

91

92

92

XV. CYCLIC FUNCTIONS.

90. Definitions of modocyclic functions.

ΙΟΙ

91. Formulæ of cyclic functions. . .

102

92. Agenda: Examples in multiples and submultiples of the

argument.

106

93. Periodicity of modocyclic, hyperbolic and circular

functions. . .

107

95.

94. Agenda: Functions of submultiples of the periods.

The inverse functions

96. Agenda: Formulæ of inverse functions.

97. Logarithmic forms of inverse cyclic functions

98. Agenda: Examples in the reduction of inverse cyclic

forms

CHAPTER V,

GRAPHICAL TRANSFORMATIONS.

XVI. ORTHOMORPHOSIS UPON THE SPHERE.

99. Affix, correspondence, morphosis.

113

XVII. PLANAR ORTHOMORPHOSIS.

ARTICLE.

104. The w-plane and the z-plane

105. The logarithmic spirals of B" non-intersecting

106. Orthomorphosis of B".

107. Isogonal relationship

108. Orthomorphosis of cosk (u + iv).

109. By confocal ellipses. .

110. By confocal hyperbolas .

III. Agenda: Problems in orthomorphosis.

CHAPTER VI.

PROPERTIES OF POLYNOMIALS.

XVIII. ROOTS OF COMPLEX QUANTITIES.

PAGE.

117

118

119

121

123

124

126

127

I12.

Definition of an nth root.

128

113. Evaluation of th roots.

128

114. Agenda: Examples in the determination of nth roots.

The fundamental theorem of algebra .

121. Agenda: Theorems concerning polynomials.

130

131

131

133

136

136

On the proof of the theorem: If the

modulus be changed from m to km the corre-

sponding base is changed from b to b1/k.

139

Art. 27, pages 43, 44. Alternative proof of the law of indices 139

The following quantitive and operational signs are used in this work and are collected here for convenience of reference.