ON THE STABILITY OF THE MOTION OF SATURN'S KINGS. 289 



CONTENTS. 



Nature of the Problem 290 



Laplace's investigation of the Equilibrium of a Ring, and its minimum density 292 



Hit proof that the plane of the Rings will follow that of Saturn's Equator that a solid uniform Ring is 



necessarily unstable 293 



Further investigation required Theory of an Irregular Solid Ring leads to the result that to ensure stability 



the irregularity must be tnormous 294 



Theory of a fluid or discontinuous Ring resolves itself into the investigation of a series of waves . . 295 



PART I. 



ON THE MOTION OF A RIGID BODY OF ANY FORM ABOUT 

 A SPHERE. 



Equations of Motion 296 



PROBLEM I. To find the conditions under which a uniform motion of the Ring is possible . . . 298 



PROBLEM II. To find the equations of the motion when slightly disturbed 299 



PROBLEM III. To reduce the three simultaneous equations of motion to the form of a single linear equation 300 

 PROBLEM IV. To determine whether the motion of the Ring is stable or unstable, by means of the relations 



of the coefficients A, B, C 301 



PROBLEM V. To find the centre of gravity, the radius of gyration, and the variations of the potential near 



the centre of a circular ring of small but variable section 302 



PROBLEM VI. To determine the condition of stability of the motion in terms of the coefficients f, g, h, which 



indicate the distribution of mass in the ring 30C 



RB60LT8. 1st, a uniform ring is unstable. 2nd, a ring varying in section according to the law of sines is 

 unstable. 3rd, a uniform ring loaded with a heavy particle may be stable, provided the mass of the 

 particle be between '815865 and '8279 of the whole. Case in which the ring is to the particle at 18 

 to 82 307 



PART II. 



ON THE MOTION OF A RING, THE PARTS OF WHICH ARE NOT 

 RIGIDLY CONNECTED. 



1. General Statement of the Problem, and limitation to a nearly uniform ring 310 



2. Notation 311 



3. Expansion of a function in terms of sines and cosines of multiples of the variable .... 311 



4. Magnitude and direction of attraction between two elements of a disturbed Ring 312 



5. Resultant attractions on any one of a ring of equal satellites disturbed in any way . . . . 313 

 Note. Calculated values of these attractions in particular cases 314 



6. Equations of motion of a satellite of the Ring, and biquadratic equation to determine the wave-velocity 315 



7. A ring of satellites may always be rendered stable by increasing the mass of the central body . . 317 



8. Relation between the number and mass of satellites and the mass of the central body necessary to 



ensure stability. S>'4352 M 2 R 318 



9. Solution of the biquadratic equation when the mass of the Ring is small ; and complete expressions 



for the motion of each satellite 319 



10. Each satellite moves (relatively to the ring) in an ellipse 321 



VOL. I. 37 



