# Book:Robert J.T. Bell/Coordinate Solid Geometry

## Robert J.T. Bell: Coordinate Solid Geometry

Published $\text {1938}$, Macmillan and Co., Limited.

Reprint of chapters I to IX of An Elementary Treatise on Coordinate Geometry of Three Dimensions from 1911.

### Contents

Publisher's Note (May, 1938)
Chapter $\text {I}$: Systems of Coordinates. The Equation to a Surface
1. Segments
2. Relations between collinear segments
3. Cartesian coordinates
4. Sign of direction of rotation
5. Cylindrical coordinates
6. Polar coordinates
7. Change of origin
8. Point dividing line in given ratio
9. The equation to a surface
10. The equation to a curve
11. Surfaces of revolution

Chapter $\text {II}$: Projections. Direction-Cosines. Direction-Ratios
12. The angles between two directed lines
13. The projection of a segment
14. Relation between a segment and a projection
15. The projection of a broken line
16. The angle between two planes
17. Relation between areas of a triangle and its projection
18. Relation between areas of a polygon and its projection
19. Realtion between areas of a closed curve and its projection
20. Direction-cosines -- definition
21, 22. Direction-cosines (axes rectangular)
23. The angle between two lines with given direction-cosines
24. Distance of a point from a line
25, 26. Direction-cosines (axes oblique)
27. The angle between two lines with given direction-cosines
28, 29, 30. Direction-ratios
31. Relation between direction-cosines and direction-ratios
32. The angle between two lines with given direction-ratios

Chapter $\text {III}$: The Plane. The Straight Line. The Volume of a Tetrahedron.
33. Forms of the equation to a plane
34, 35. The general equation to a plane
36. The plane through three points
37. The distance of a point from a plane
38. The planes bisecting the angles between two given planes
39. The equations to a straight line
40. Symmetrical form of equations
41. The line through two given points
42. The direction-ratios found from the equations
43. Constants in the equations to a line
44. The plane and the straight line
45. The intersection of three planes
46. Lines intersecting two given lines
47. Lines intersecting three given lines
48. The condition that two given lines be coplanar
49. The shortest distance between two given lines
50. Problems relating to two given non-intersecting lines
51. The volume of a tetrahedron

Chapter $\text {IV}$: Change of Axes
52. Formulae of transformation (rectangular axes)
53. Relations between the directin-cosines of three mutually perpendicular lines
54. Transformation to examine the section of a given surface by a given plane
55. Formulae of transformation (oblique axes}
Examples $\text {I}$

Chapter $\text {V}$: The Sphere
56. The equation to a sphere
57. Tangents and tangent plane to a sphere
58. The radical plane of two spheres
Examples $\text {II}$

Chapter $\text {VI}$: The Cone
59. The equation to a cone
60. The angle between the lines in which a plane cuts a cone
61. The condition of tangency of a plane and a cone
62. The condition that a cone has three mutually perpendicular generators
63. The equation to a cone with a given base
Examples $\text {III}$

Chapter $\text {VII}$: The Central Conicoids. The Cone. The Paraboloids
64. The equation to a central conicoid
65. Diametral planes and conjugate diameters
66. Points of intersection of a line and a conicoid
67. Tangents and tangent planes
68. Condition that a plane should touch a conicoid
69. The polar plane
70. Polar lines
71. Section with a given centre
72. Locus of the mid-points of a system of parallel chords
73. The enveloping cone
74. The enveloping cylinder
75. The normals
76. The normals from a given point
77. Conjugate diameters and diametral planes
78. Properties of the cone
79. The equation of a paraboloid
80. Conjugate diametral planes
81. Diameters
82. Tangent planes
83. Diametral planes
84. The normals
Examples $\text {IV}$

Chapter $\text {VIII}$: The Axes of Plane Sections. Circular Sections.
85. The determination of axes
86. Axes of a central section of a central conicoid
87. Axes of any section of a central conicoid
88. Axes of a section of a paraboloid
89. The determination of circular sections
90. Circular sections of the ellipsoid
91. Any two circular sections from opposite systems lie on a sphere
92. Circular sections of the hyprboloids
93. Circular sections of the general central conicoid
94. Circular sections of the paraboloids
95. Umbilics
Examples $\text {V}$

Chapter $\text {IX}$: Generating Lines
96. Ruled surfaces
97. The section of a surface by a tangent plane
98. Line meeting conicoid in three points of a generator
99. Conditions that a line should be a generator
100. System of generators of a hyperboloid
101. Generators of same system do not intersect
102. Generators of opposite systems intersect
103. Locus of points of intersection of perpendicular generators
104. The projections of generators
105. Along a generator $\theta \pm \phi$ is constant
106. The systems of generators of the hyperbolic paraboloid
107. Conicoids through three given lines
108. General equation to conicoid through two given lines
109. The equation to the conicoid through three given lines
110, 111. The straight lines which meet four given lines
112. The equation to a hyperboloid when generators are coordinate axes
113. Properties of a given generator
114. The central point and parameter of distribution
Examples $\text {VI}$

APPENDIX
MISCELLANEOUS EXAMPLES $\text {I}$.
MISCELLANEOUS EXAMPLES $\text {II}$.
INDEX