WCP2002

[1]

George Darwin's Notes

1. It is probable that An symmetrical annular system of bodies might move revolve in a circle with or without a central body. Such a system wold be unstable. If the bodies are of unequal masses and not symetrically disposed[?] the[?] break up of the system would probably be more rapid[?] than in the ideal case of symmetry.¹ According to Maxwell's investigations Saturn's rings are also unstable, but the instability is excessively slow, so that in ordinary parlance one might describe the system is[as?] possessing temporary stability. If the central mass were absent, or if it was not on any times as heavy as the whole [illeg]ing the instability would certainly be rapid.

2. The conditions which will ensure stability in a globular star cluster are wholly unknown. The problem of three bodies is only in its infancy, and à fortiori[?] the problem of the relative motion of a large number. The only thing which can prevent the falling together of stars is their relative motion. Judging by analogy it would seem probable that the bodies must all circulate one way in order to ensure any kind of stability; [2 words crossed out illeg.] this may be expressed by saying that they all revolve about their common centre of inertia. This conclusion is merely an expression of probability.

In any case the investigations of Poincaré² seem to show that any stability which the system can possess can only be of the temporary kind adverted to under (1); ultimately the system must ^{almost certainly} be unstable.

I cannot answer 2.a., but I conjecture that stability would be best ensured by approximate identity of the planes of the orbits. A predominant central body would no doubt contribute to stability. Indeed it seems open to doubt whether even temporary stability could be secured by without such a predominant central body. I conjecture doubtfully that the tendency would be towards the concentration either towards a single mass or to two revolving about one another[.]

3 & 4 I have no opinion as to these.