MR. F. W. DYSON ON THE POTENTIAL OF AN ANCHOR RING. 1043 
The special cases of one ring pursuing another of equal strength, of the direct 
approach of a ring towards a fixed plane, and of the motion of a ring over a spherical 
obstacle are considered in detail. 
Section I. 
§§ 1-4. Solution of Laplace’s equation inside an anchor ring. 
§§ 5-7. Potential of a solid anchor ring at internal points. 
Exhaustion of potential energy. 
§§ 8-9. Potential of a distribution of matter on the surface of a ring, at all 
internal points, and at external points near the ring. 
Section II. 
§§ 10-13. Potential and exhaustion of potential energy of a ring whose cross- 
section is R = a (1 + S /d„ cos ny). 
§§ 14-15. Small fluted oscillations of a gravitating fluid ring. 
§§ 16-19. Exhaustion of potential energy of a ring which has beaded waves on it. 
§ 20. Such waves render the ring unstable. 
§§ 21-22. Exhaustion of potential energy of a ring whose central circle is deformed. 
§ 23. Such deformation does not produce instability. 
Section III. 
§§ 24-26. Potential of a ring, whose cross-section is elliptic, at external points. 
§ 27. When the elliptic section is very flat. 
§ 28. Application to Saturn. 
Section IV. 
§ 29. Steady motion of a single thick vortex ring. 
§§ 30-34. The stream-line function. 
§§ 35-36. Determination of the velocity and the form of the cross-section. 
§ 37. Small fluted oscillations. 
Section V. 
§ 38. Stream-line function for any number of fine vortex rings on the same 
axis. 
§ 39. The equations of motion of the rings. 
§§ 40-42. The integrals of momentum and energy. 
§§ 43-47. Special cases of two rings. 
6 R 2 
