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xii

PROBLEM

46. To describe a parabola with given axis and to pass through two

given points

47. To describe a parabola with given axis, to pass through a given

point and to touch a given line

48.

CONTENTS.

51.

To describe a parabola to pass through three points and the

axis to be in a given direction

53.

To describe a parabola with given axis and to touch two given

lines

54.

49.

To describe a parabola to touch two given lines at given points

50. To describe a parabola to touch three given lines, one of them

at a given point .

52. To describe a parabola to touch three given lines and the axis

to be in a given direction

•

To describe a parabola to pass through two given points and to

touch two given lines

To describe a parabola to pass through three given points and to

touch a given line

55. To describe a parabola to pass through a given point and to

touch three given lines

56. To describe a parabola to pass through four given points .

57. To describe a parabola to touch four given lines

58. To determine the centre of curvature at any point of a given

parabola

59. To describe a parabola to touch two given circles, the axis being

the line joining their centres

CHAPTER IV.

THE ELLIPSE.

60.

To describe an ellipse with given axes (three methods)

Tangent and normal

61. To describe approximately by means of circular arcs an ellipse

having given axes (two methods)

Sundry properties of the ellipse.

62. To determine the axes of an ellipse from a given pair of con-

jugate diameters

63. To describe an ellipse with given conjugate diameters (four

methods)

64. To describe an ellipse with a given axis and to pass through a

given point

65.

66.

67.

68.

69. To describe an ellipse, the centre, the directions of a pair of

conjugate diameters, a tangent and a point on the curve

being given

70. To describe an ellipse, the centre, two tangents and a point on

the curve being given

71.

To describe an ellipse, the centre and three tangents being given

72. To describe an ellipse, the centre, two points on the curve and

a tangent being given

74.

73. To describe an ellipse, the centre and three points on the curve

75.

76.

77.

To describe an ellipse with a given axis and to touch a given line

To describe an ellipse, the directions of a pair of conjugate dia-

meters, a tangent and its point of contact being given

To describe an ellipse, the centre, two points on the curve and the directions of a pair of conjugate diameters being given

To describe an ellipse, the centre, the direction of the major

axis and two tangents being given

78.

82.

83.

To describe an ellipse, the foci and a point on the curve being

given

To describe an ellipse, the foci and a tangent to the curve being given.

To describe an ellipse, a focus, a tangent with its point of

contact and a second point on the curve being given.

To describe an ellipse, a focus, a tangent and two points on the

curve being given

To describe an ellipse, a focus, a point on the curve and two tangents being given

79. To describe an ellipse, a focus and three tangents being given

80. To describe an ellipse, a focus and three points being given

81. To describe an ellipse, two tangents with their points of contact

and a third point being given

To describe an ellipse, two tangents and three points being given

To describe an ellipse, three tangents and two points being given

PAGE

111

114

115

116

ib.

118

119

121

122

123

124

125

126

127

129

131

133

134

xiv

84.

85.

86.

87.

89.

91.

92,

To describe an ellipse, five tangents being given

To describe an ellipse, four tangents and a point being given

To describe an ellipse, five points being given

To describe an ellipse, four points and a tangent being given

Pole and Polar

93.

Harmonic Properties.

88. To determine the centre of curvature at any point of a given

ellipse

CHAPTER V.

THE HYPERBOLA.

90. To describe an hyperbola, an asymptote, focus and a point on the curve being given .

To describe an hyperbola, the foci and a vertex, the vertices and a focus, or the axes, being given

To describe an hyperbola, an asymptote, focus and tangent

To describe an hyperbola, an asymptote, directrix and a point

To describe an hyperbola, the asymptotes and a point being

94. To describe an hyperbola, the asymptotes and a tangent being

given.

Sundry properties of hyperbola .

95. To describe an hyperbola, transverse axis and a point being

96. To describe an hyperbola, transverse axis and tangent being

97. To describe an hyperbola, a pair of conjugate diameters being

136

137

138

139

140

141

145

153

157

158

159

160

166

98. To describe an hyperbola, the centre, directions of a pair of con-

jugate diameters and two points being given

99. To describe an hyperbola, the centre, directions of a pair of con-

jugate diameters, a tangent and a point being given

100. To describe an hyperbola, the centre, two tangents and a point

101.

102.

To describe an hyperbola, the centre, a tangent and two points

103. To describe an hyperbola, the centre and three points being given

104. To describe an hyperbola, the foci and a point on the curve being

To describe an hyperbola, the centre and three tangents being

105. To describe an hyperbola, the foci and a tangent being given

106. To describe an hyperbola, a focus, tangent with its point of contact and a second point on the curve being given.

107. To describe an hyperbola, a focus, a tangent and two points

109.

110.

108. To describe an hyperbola, a focus, two tangents and a point

111.

To describe an hyperbola, a focus and three tangents being given

To describe an hyperbola, a focus and three points on the curve

To describe an hyperbola, two tangents with their points of

contact and a third point on the curve being given

112. To describe an hyperbola, three tangents and two points on the

113. To describe an hyperbola, two tangents and three points on the

114. To describe an hyperbola, five tangents being given

115. To describe an hyperbola, five points on the curve being given .

116. To describe an hyperbola, four tangents and one point being

117. To describe an hyperbola, four points and one tangent being

118. To determine the centre of curvature at any point of a given

hyperbola

168

170

171

172

173

174

175

176

178

179

181

182

183

184

185

186

187

188

119. To find the polar reciprocal of one circle with regard to another

CHAPTER VIII.

ANHARMONIC RATIO.

120. Given the anharmonic ratio of four points, and the position of

three of them, to determine the fourth

191-195

121. Given any number of points on a straight line, and three points

on a second line corresponding to a given three on the first,

to complete the homographic division of the second line

122. Given a pencil of rays, and three rays of a second pencil corresponding to a given three of the first, to complete the second so that the two shall be homographic

123. Given two homographic ranges in the same straight line, to

determine the double points

126. Given two straight lines

draw Rab so that

124. Given two pairs of conjugate points and a fifth point of the

involution, to determine its conjugate

125. Given A, a and B, b in a straight line, to find in the same line a

point M such that

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