CONTENTS. vil 



spectively, 51. Central force in an ellipse when the centre 

 of force is the centre of the ellipse, 53. Converse of the pro 

 position, 54-. Equality of periodic times in concentric similar 

 curves, when the law of the force is as the distance, ib. Con 

 sequence of thejsun being in the centre of the system, 55. 



(SECTION III. Principia.) Law of forces when the centre of forces 

 is in the focus of the curve, ib. General theorem that in each 

 of the three conic sections the law is the inverse square of the 

 distance, ib. Converse of the proposition proved, 57. J. Ber- 

 nouiili s objection to Sir Isaac Newton s proof, 58. Shown to 

 be groundless, ib. His objection to Herrman s demonstration, 



59. Refuted, ib. Motion in concentric conic sections, the 

 centre of forces being in the focus, ib. Demoivre s theorem, 



60. Demonstration of Kepler s law of sesquiplicate ratio 

 generally, 6l. Inverse problem of finding the orbit from the 

 force being given, ib. Determination of the nature of the 

 orbit from the forces, 62. Sir Isaac Newton s observations 

 on the investigation of disturbing forces, ib. Anticipates La- 

 grange s investigation, 63. note. Importance of Perpendicular 

 to the Tangent and Radius of Curvature in all these inquiries, 63. 



i. (SECTIONS IV. V. Principia.) General observations on these 

 sections, 64. Illustration of their use in Physical Astronomy, 

 65. Further illustration from their application to the problems 

 on comets, ib. Comparison of theory with observation by 

 Newton, 66. By Halley, 6?. Comets of 1680, 1665, 1682, 

 1683, ib. General remarks on the importance of these sections, 

 68. 



iii. Motion (1) in given conic sections, (2) in straight lines, 

 ascending or descending. 



iii. (1.) (SECTION VI. Principia.} Method of determining 

 the place of a body in a given trajectory, being a conic section, at 

 any given time, 69. Solution for the parabola, 70. Method 

 conversely of finding the time, the places being given, 71. 

 Solution for the ellipse, or Kepler s problem, ib. Difficulty 

 of the problem, 72. Sir Isaac Newton s proof that no oval is 

 quadrable, ib. Class of curves returning into themselves and 

 quadrable, beside the class mentioned by him of ovals connected 

 with infinite branches, 73. Demonstration respecting the 

 ellipse, 74. Observations, ib. Sir Isaac Newton s solution 

 of Kepler s problem indirectly by the cycloidal, ib. Another 

 solution directly by a cycloidal curve, 75. Astronomical No 

 menclature, 76. 



ii (2.) (SECTION VII. Principia. Motion ascending and 

 descending in straight lines, ib. Determination of times of 

 descent and ascent, ib. Determination of velocities in case of 

 parabolic lines, 77. Time of moon falling to the earth, 79- 



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