290 ON THE STABILITY OP THE MOTION OP SATURN'S RINGS. 



11. Each satellite motet absolutely in tpaet in a curve which it nearly an ellipse for the large valuet 



of n, and a spiral of many nearly circular coil* when n it mall 321 



13. Tke form of the ring at a given inttant it a tenet of undulations 322 



13. These undulations travel round the ring with velocity - - relative to the ring, and absolutely 323 



14. General Solution of the Problem Given the porition and motion of every satellite at any one time, 

 to calculate the petition and motion of any tateUite at any other time, provided that the condition 



of ttability it fulfilled 323 



15. Calculation of the efect of a periodic external disturbing force 326 



16. Treatment of ditturbing foroet in general 328 



17. Theory of free waww and forced waves 329 



18. Motion of the ring when the condition* of ttability are not fulfilled. Two different ways in which. 



the ring may be broken up 330 



19. Motion of a ring of unequal tatettitet 335 



20. Motion of a ring composed of a cloud of scattered particle* 336 



21. Calculation of the forces arising from the displacements of such a system 337 



22. Application to the cote of a ring of this kind. The mean density must be excessively small, which 



it inconsistent with its moving as a whole 338 



23. On the forces arising from inequalities in a thin stratum of gravitating incompressible fluid of 

 indefinite extent 338 



24. Application to the cote of a flattened fluid ring, moving with uniform angular velocity. Such a 



ring will be broken up into portions which may continue to revolve a* a ring of tatettitet . . . 344 



ON THE MUTUAL PERTURBATIONS OF TWO RINGS. 



25. Application of the general theory of free and forced waves 345 



26. To determine the attractions between the rings 346 



27. To form the equation* of motion 348 



28. Method of determining the reaction of the forced u-art on the free wave which produced it . . 349 



29. C<i*et in which the perturbations increase indefinitely 351 



30. Application to the theory of an indefinite number of concentric rings 352 



31. On the efect of long-continued disturbances on a system of rings - . . . 352 



32. On the efect of collision* among the parts of a revolving system 354 



33. On the effect of internal friction in a fluid ring 354 



Recapitulation of the Theory of the Motion of a Rigid King. Reasons for rejecting the hypothesis of 



rigidity 356 



Recapitulation of the Theory of a Ring of Equal Satellites 360 



Description of a working model shewing the motions of such a system 363 



Theory of Rings of various constitution* 367 



Mutual action of Two Rings 370 



Catt of many concentric Rings, <k<:. 371 



General Conclusions 372 



APPENDIX. Extract of a letter from Professor W. Thomson, of Glasgow, giving a solution of t/ie Pro- 

 blem of a Rigid Ring 374 



THERE are some questions in Astronomy, to which we are attracted rather 

 on account of their peculiarity, as the possible illustration of some unknown 

 principle, than from any direct advantage which their solution would afford to 



