Vlll CONTENTS. 



Analogy of the case of planets falling into the sun, to the 

 structure of bees cells, ib. note. General solution of the pro 

 blem for all kinds of centripetal force and orbit, 79- 



iv. (SECTION VIII. Principia.) Observations upon the general 

 inverse problem of centripetal forces, or finding the orbit, the 

 force being given, 80. Sir Isaac Newton s solution, though geo 

 metrical, is less synthetical than usual, 82. Determination of 

 the trajectory generally by the method of quadratures, ib. Re 

 marks on that method, 85. The subject illustrated in the case 

 of the inverse cube of the distance, ib. Another solution given 

 by a polar equation, 86. Conclusion of the subject of centri 

 petal forces in fixed orbits, and round an immoveable centre, ib. 



Of motion in moveable orbits divided into two heads, 87. i. 

 When the orbit and centre are in the same plane. ii. When 

 the orbit s plane is eccentric. 



i. (SECTION IX. Principia.) Determination of the motion of 

 the apsides, ib. Proportion of force to distance, which make 

 the axis or apsides advance and retire respectively, 88. Deter 

 mination of motion of apsides from the force and conversely, 

 89. Gravitation the only force by which the line of apsides 

 can coincide with the fixed axis, 90. Motion of the apsides 

 with different centripetal forces, ib. Application of the theory 

 to the motion of the moon s apsides, 91. To the motion of the 

 earth s apsides, 92. Sir Isaac Newton did not reconcile the 

 theory with observation, as regards the moon, ib. Misstate- 

 ment of Bailly on this subject, ib. History of the question 

 respecting the agreement of the theory with the observation, 



93. Euler, D Alembert, Clairaut, ib. Clairaut s error, and 

 his discovery of the agreement between the theory and fact, 



94. Laplace s solution and discoveries, ib Reference to 

 the papers of the three mathematicians on the problem of these 

 bodies, ib. note. Bailly s further erroneous statement respecting 

 Sir Isaac Newton, 95. Proof of that error, ib. General 

 opinion of Bailly on the Newtonian lunar theory erroneous, 96. 

 Testimony of Laplace, 97- Error of Laplace respecting Sir 

 Isaac Newton s assumption as to the perigeal motion, ib. 



ii. (SECTION X. Principia.) Determination of trajectories in a given 

 plane, when the centre is out of that plane, 98. Of trajectories 

 on a curve surface, 100. Example of the circle and cylinder, 

 ib. Motion of pendulums, 101. Properties of hypercycloids, 



and hypocycloids, ib. Isochronism of the cycloid, 102. General 

 solution for all curves by the evolutes, ib. Peculiarity of cycloid 

 and logarithmic spiral in being their own evolutes, 103. Reason 

 why Sir Isaac Newton took the case of hypercycloids and hypo- 

 cycloids, and not cycloids, ib Measurement of gravity by the 

 pendulum, deduced from these propositions, ib. Conclusion 



