One of the theorems of a paper which Professor Kelland did me the honour to read to the Society, in March 1865, opens up a field of geometrical investigation so interesting and fertile, that I venture to ask attention to some of the results of a partial examination of it in the following series of propositions. I think it right to explain, that I do not venture to expect attention to them on account of any importance attaching to them individually, but on account of their number and somewhat elegant relations. Considered individually, they may be of little importance, having no claim to rank, so to speak, among propositions of a planetary magnitude. But a system of moons, however diminutive, may become interesting if they present elegant relations among their mean motions and longitudes; and an orbit that would be grudged to a pigmy planet may be willingly accorded to a host of planetoids. If this is still too exalted language in which to speak of the following results, I can at least confidently affirm that they indicate a direction in which an analyst of very moderate attainments may easily discover for himself a shower of meteors.