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

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## Robert J.T. Bell:

## 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.

### Subject Matter

### 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